4th March 2019, version of 13 November 2023, 13:44.
Martín Hötzel Escardó, School of Computer Science, University of Birmingham, UK.
Abstract. We introduce Voevodsky’s univalent foundations and univalent mathematics, and explain how to develop them with the computer system Agda, which is based on Martin-Löf type theory. Agda allows us to write mathematical definitions, constructions, theorems and proofs, for example in number theory, analysis, group theory, topology, category theory or programming language theory, checking them for logical and mathematical correctness.
Agda is a constructive mathematical system by default, which amounts to saying that it can also be considered as a programming language for manipulating mathematical objects. But we can assume the axiom of choice or the principle of excluded middle for pieces of mathematics that require them, at the cost of losing the implicit programming-language character of the system. For a fully constructive development of univalent mathematics in Agda, we would need to use its new cubical flavour, and we hope these notes provide a base for researchers interested in learning cubical type theory and cubical Agda as the next step.
Compared to most expositions of the subject, we work with explicit universe levels.
We also fully discuss and emphasize that non-constructive classical axioms can be assumed consistently in univalent mathematics.
Keywords. Univalent mathematics. Univalent foundations. Univalent
type theory. Univalence axiom. ∞-Groupoid. Homotopy type. Type
theory. Homotopy type theory. HoTT/UF. Intensional Martin-Löf type
theory. Dependent type theory. Identity type. Type
universe. Constructive mathematics. Agda. Cubical type
theory. Cubical Agda. Computer-verified mathematics.
About this document. This is a set of so-called literate Agda files, with the formal, verified, mathematical development within code environments, and the usual mathematical discussion outside them. Most of this file is not Agda code, and is in markdown format, and the html web page is generated automatically from it using Agda and other tools. Github issues or pull requests by students to fix typos or mistakes and clarify ambiguities are welcome. There is also a pdf version with internal links to sections and Agda definitions, which is automatically generated from the html version. And a pdf version is also available at the arxiv, updated from time to time.
These notes were originally developed for the Midlands Graduate School 2019. They will evolve for a while.
A univalent type theory is the underlying formal system for a foundation of univalent mathematics as conceived by Voevodsky.
In the same way as there isn’t just one set theory (we have e.g. ZFC and NBG among others), there isn’t just one univalent type theory (we have e.g. the underlying type theory used in UniMath, HoTT-book type theory, and cubical type theory, among others, and more are expected to come in the foreseeable future before the foundations of univalent mathematics stabilize).
The salient differences between univalent mathematics and traditional, set-based mathematics may be shocking at first sight:
The kinds of objects we take as basic.
ℕ of natural numbers is a set, and this is a theorem, not a definition.1-groupoid, automatically.2-groupoid, again automatically.The treatment of logic.
-1-groupoids, with at most one element.The treatment of equality.
x ＝ y is a type, called the identity type, which is not necessarily a truth value.x and y are identified.ℕ, as there is at most one way for two natural numbers to be equal.1-groupoid, amounting to the type of equivalences of categories, again automatically.The important word in the above description of univalent foundations is automatic. For example, we don’t define equality of monoids to be isomorphism. Instead, we define the collection of monoids as the large type of small types that are sets, equipped with a binary multiplication operation and a unit satisfying associativity of multiplication and neutrality of the unit in the usual way, and then we prove that the native notion of equality that comes with univalent type theory (inherited from Martin-Löf type theory) happens to coincide with the notion of monoid isomorphism. Largeness and smallness are taken as relative concepts, with type universes incorporated in the theory to account for the size distinction.
In particular, properties of monoids are automatically invariant under isomorphism, properties of categories are automatically invariant under equivalence, and so on.
Voevodsky’s way to achieve this is to start with a Martin-Löf type theory (MLTT), including identity types and type universes, and postulate a single axiom, named univalence. This axiom stipulates a canonical bijection between type equivalences (in a suitable sense defined by Voevodsky in type theory) and type identifications (in the original sense of Martin-Löf’s identity type). Voevodsky’s notion of type equivalence, formulated in MLTT, is a refinement of the notion of isomorphism, which works uniformly for all higher groupoids, i.e. types.
In particular, Voevodsky didn’t design a new type theory, but instead gave an axiom for an existing type theory (or any of a family of possible type theories, to be more precise).
The main technical contributions in type theory by Voevodsky are:
Univalent mathematics begins within MLTT with (4) and (5) before we postulate univalence. In fact, as the reader will see, we will do a fair amount of univalent mathematics before we formulate or assume the univalence axiom.
All of (4)-(6) crucially rely on Martin-Löf’s identity type. Initially, Voevodsky thought that a new concept would be needed in the type theory to achieve (4)-(6) and hence (1) and (3) above. But he eventually discovered that Martin-Löf’s identity type is precisely what he needed.
It may be considered somewhat miraculous that the addition of the univalence axiom alone to MLTT can achieve (1) and (3). Martin-Löf type theory was designed to achieve (2), and, regarding (1), types were imagined/conceived as sets (and even named sets in some of the original expositions by Martin-Löf), and, regarding (3), the identity type was imagined/conceived as having at most one element, even if MLTT cannot prove or disprove this statement, as was eventually shown by Hofmann and Streicher with their groupoid model of types in the early 1990’s.
Another important aspect of univalent mathematics is the presence of explicit mechanisms for distinguishing
which are common place in current mathematical practice (e.g. cartesian closedness of a category is a property in some sense (up to isomorphism), whereas monoidal closedness is given structure).
In summary, univalent mathematics is characterized by (1)-(8) and not by the univalence axiom alone. In fact, half of these notes begin without the univalence axiom.
Lastly, univalent type theories don’t assume the axiom of choice or the principle of excluded middle, and so in some sense they are constructive by default. But we emphasize that these two axioms are consistent and hence can be safely used as assumptions. However, virtually all theorems of univalent mathematics, e.g. in UniMath, have been proved without assuming them, with natural mathematical arguments. The formulations of these principles in univalent mathematics differ from their traditional formulations in MLTT, and hence we sometimes refer to them as the univalent principle of excluded middle and the univalent axiom of choice.
In these notes we will explore the above ideas, using Agda to write MLTT definitions, constructions, theorems and proofs, with univalence as an explicit assumption each time it is needed. We will have a further assumption, the existence of certain subsingleton (or propositional, or truth-value) truncations in order to be able to deal with the distinction between property and data, and in particular with the distinction between designated and unspecified existence (for example to be able to define the notions of image of a function and of surjective function).
We will not assume univalence and truncation globally, so that the students can see clearly when they are or are not needed. In fact, the foundational definitions, constructions, theorems and proofs of univalent mathematics don’t require univalence or propositional truncation, and so can be developed in a version of the original Martin-Löf type theories, and this is what happens in these notes, and what Voevodsky did in his brilliant original development in the computer system Coq. Our use of Agda, rather than Coq, is a personal matter of taste only, and the students are encouraged to learn Coq, too.
Univalent type theory is often called homotopy type theory. Here we are following Voevodsky, who coined the phrases univalent foundations and univalent mathematics. We regard the terminology homotopy type theory as probably more appropriate for referring to the synthetic development of homotopy theory within univalent mathematics, for which we refer the reader to the HoTT book.
However, the terminology homotopy type theory is also used as a synonym for univalent type theory, not only because univalent type theory has a model in homotopy types (as defined in homotopy theory), but also because, without considering models, types do behave like homotopy types, automatically. We will not discuss how to do homotopy theory using univalent type theory in these notes. We refer the reader to the HoTT book as a starting point.
A common compromise is to refer to the subject as HoTT/UF.
ncatlab references.In particular, it is recommended to read the concluding notes for each chapter in the HoTT book for discussion of original sources. Moreover, the whole HoTT book is a recommended complementary reading for this course.
And, after the reader has gained enough experience:
Regarding the computer language Agda, we recommend the following as starting points:
Regarding the genesis of the subject:
Voevodsky says that he was influenced by Makkai’s thinking:
An important foundational reference, by Steve Awodey and Michael A. Warren, is
Additional expository material:
More references as clickable links are given in the course of the notes.
We also have an Agda development of univalent foundations which is applied to work on injective types, compact (or searchable) types, compact ordinals and more.
This is intended as an introductory graduate course. We include what we regard as the essence of univalent foundations and univalent mathematics, but we are certainly omitting important material that is needed to do univalent mathematics in practice, and the readers who wish to practice univalent mathematics should consult the above references.
𝓤,𝓥,𝓦𝟙𝟘ℕ of natural numbers_+_Σ typesΠ typesId, also written _＝_There are two views:
Agda is a dependently-typed functional programming language.
Agda is a language for defining mathematical notions (e.g. group or topological space), formulating constructions to be performed (e.g. a type of real numbers, a group structure on the integers, a topology on the reals), formulating theorems (e.g. a certain construction is indeed a group structure, there are infinitely many primes), and proving theorems (e.g. the infinitude of the collection of primes with Euclid’s argument).
This doesn’t mean that Agda has two sets of features, one for (1) and the other for (2). The same set of features account simultaneously for (1) and (2). Programs are mathematical constructions that happen not to use non-constructive principles such as excluded middle.
In these notes we study a minimal univalent type theory and do mathematics with it using a minimal subset of the computer language Agda as a vehicle.
Agda allows one to construct proofs interactively, but we will not discuss how to do this in these notes. Agda is not an automatic theorem prover. We have to come up with our own proofs, which Agda checks for correctness. We do get some form of interactive help to input our proofs and render them as formal objects.
Before embarking into a full definition of our Martin-Löf type theory in Agda, we summarize the particular Martin-Löf type theory that we will consider, by naming the concepts that we will include. We will have:
An empty type 𝟘.
A one-element type 𝟙.
A type of ℕ natural numbers.
Type formers + (binary sum),
Π (product),
Σ (sum),
Id (identity type).
Universes (types of types), ranged
over by 𝓤,𝓥,𝓦.
This is enough to do number theory, analysis, group theory, topology, category theory and more.
spartan /ˈspɑːt(ə)n/ adjective:
showing or characterized by austerity or a lack of comfort or
luxury.
We will also be rather spartan with the subset of Agda that we choose to discuss. Many things we do here can be written in more concise ways using more advanced features. Here we introduce a minimal subset of Agda where everything in our spartan MLTT can be expressed.
We don’t use any Agda library. For pedagogical purposes, we start from scratch, and here are our first two lines of code:
{-# OPTIONS --without-K --exact-split --safe --auto-inline #-} module HoTT-UF-Agda where
The option --without-K disables Streicher’s K axiom, which we don’t
want for univalent mathematics.
The option --exact-split makes Agda to only accept definitions
with the equality sign “=” that behave like so-called
judgmental or definitional equalities.
The option --safe disables features that may make Agda
inconsistent,
such as --type-in-type, postulates and more.
Every Agda file is a module. These lecture notes are a set of Agda files, which are converted to html by Agda after it successfully checks the mathematical development for correctness.
The Agda code in these notes has syntax highlighting and links (in the html and pdf versions), so that we can navigate to the definition of a name or symbol by clicking at it.
A universe 𝓤 is a type of types.
One use of universes is to define families of types indexed by a
type X as functions X → 𝓤.
Such a function is sometimes seen
as a property of elements of X.
Another use of universes, as we shall see, is to define types of mathematical structures, such as monoids, groups, topological spaces, categories etc.
Sometimes we need more than one universe. For example, the type of groups in a universe lives in a bigger universe, and given a category in one universe, its presheaf category also lives in a larger universe.
We will work with a tower of type universes
𝓤₀, 𝓤₁, 𝓤₂, 𝓤₃, ...
In practice, the first one, two or three universes suffice, but it will be easier to formulate definitions, constructions and theorems in full generality, to avoid making universe choices before knowing how they are going to be applied.
These are actually universe names (also called levels, not to be confused with hlevels). We reference the universes themselves by a deliberately almost-invisible superscript dot. For example, we will have
𝟙 : 𝓤₀ ̇
where 𝟙 is the one-point type to be defined shortly. We will sometimes
omit this superscript in our discussions, but are forced to write it
in Agda code. We have that the universe 𝓤₀ is a type in the universe
𝓤₁, which is a type in the universe 𝓤₂ and so on.
𝓤₀ ̇: 𝓤₁ ̇
𝓤₁ ̇: 𝓤₂ ̇
𝓤₂ ̇: 𝓤₃ ̇
⋮
The assumption that 𝓤₀ : 𝓤₀ or that any universe is in itself or a
smaller universe gives rise to a
contradiction,
similar to Russell’s
Paradox.
Given a universe 𝓤, we denote by
𝓤 ⁺
its successor universe. For example, if 𝓤 is 𝓤₀ then 𝓤 ⁺ is
𝓤₁. According to the above discussion, we have
𝓤 ̇ : 𝓤 ⁺ ̇
The least upper bound of two universes 𝓤 and 𝓥 is written
𝓤 ⊔ 𝓥.
For example, if 𝓤 is 𝓤₀ and 𝓥 is 𝓤₁, then 𝓤 ⊔ 𝓥 is 𝓤₁.
We now bring our notation for universes by importing our Agda file
Universes. The Agda keyword
open
asks to make all definitions in the file Universe visible in our
file here.
open import Universes public
The keyword public makes the contents of the file Universes
available to importers of our module HoTT-UF-Agda.
We will refer to universes by letters 𝓤,𝓥,𝓦,𝓣:
variable 𝓤 𝓥 𝓦 𝓣 : Universe
In some type theories, the universes are cumulative “on the nose”, in
the sense that from X : 𝓤 we derive that X : 𝓤 ⊔ 𝓥. We will
instead have an embedding 𝓤 → 𝓤 ⊔
𝓥 of universes into larger universes.
𝟙We place it in the first universe, and we name its unique element
“⋆”. We use the data declaration in Agda to introduce it:
data 𝟙 : 𝓤₀ ̇ where ⋆ : 𝟙
It is important that the point ⋆ lives in the type 𝟙 and in no other
type. There isn’t dual citizenship in our type theory. When we create
a type, we also create freshly new elements for it, in this case
“⋆”. (However, Agda has a limited form of overloading, which allows
us to sometimes use the same name for different things.)
Next we want to give a mechanism to prove that all points of the
type 𝟙 satisfy a given property A.
The property is a function A : 𝟙 → 𝓤 for some universe 𝓤.
The type A(x), which we will write simply A x, doesn’t need
to be a truth value.
It can be any type. We will see examples shortly.
In MLTT, mathematical statements are types, such as
Π A : 𝟙 → 𝓤, A ⋆ → Π x : 𝟙, A x.
We read this in natural language as “for any given property A
of elements of the type 𝟙, if A ⋆ holds, then it follows that
A x holds for all x : 𝟙”.
In Agda, the above type is written as
(A : 𝟙 → 𝓤 ̇ ) → A ⋆ → (x : 𝟙) → A x.
This is the type of functions with three arguments A : 𝟙 → 𝓤 ̇
and a : A ⋆ and x : 𝟙, with values in the type A x.
A proof of a mathematical statement rendered as a type is a
construction of an element of the type. In our example, we have
to construct a function, which we will name 𝟙-induction.
We do this as follows in Agda, where we first declare the type of the
function 𝟙-induction with “:” and then define the function by an
equation:
𝟙-induction : (A : 𝟙 → 𝓤 ̇ ) → A ⋆ → (x : 𝟙) → A x 𝟙-induction A a ⋆ = a
The universe 𝓤 is arbitrary, and Agda knows 𝓤 is a universe variable because we said so above.
Notice that we supply A and a as arbitrary arguments, but instead of
an arbitrary x : 𝟙 we have written “⋆”. Agda accepts this because it
knows from the definition of 𝟙 that “⋆” is the only element of the
type 𝟙. This mechanism is called pattern matching.
A particular case of 𝟙-induction occurs when the family A is constant
with value B, which can be written variously as
A x = B
or
A = λ (x : 𝟙) → B,
or
A = λ x → B
if we want Agda to figure out the type of x by itself, or
A = λ _ → B
if we don’t want to name the argument of A because it
is not used. In usual mathematical practice, such a lambda expression is often
written
x ↦ B(xis mapped toB)
so that the above amount to A = (x ↦ B).
Given a type B and a point b : B, we construct the function 𝟙 → B
that maps any given x : 𝟙 to b.
𝟙-recursion : (B : 𝓤 ̇ ) → B → (𝟙 → B) 𝟙-recursion B b x = 𝟙-induction (λ _ → B) b x
The above expression B → (𝟙 → B) can be written as B → 𝟙 → B,
omitting the brackets, as the function-type formation symbol → is
taken to be right associative.
Not all types have to be seen as mathematical statements (for example
the type ℕ of natural numbers defined below). But the above definition
has a dual interpretation as a mathematical function, and as the
statement “B implies (true implies B)” where 𝟙 is the type encoding
the truth value true.
The unique function to 𝟙 will be named !𝟙. We define two versions
to illustrate implicit
arguments,
which correspond in mathematics to “subscripts that are omitted when
the reader can safely infer them”, as for example for the identity
function of a set or the identity arrow of an object of a category.
!𝟙' : (X : 𝓤 ̇ ) → X → 𝟙 !𝟙' X x = ⋆ !𝟙 : {X : 𝓤 ̇ } → X → 𝟙 !𝟙 x = ⋆
This means that when we write
!𝟙 x
we have to recover the (uniquely determined) missing type X with x : X
“from the context”. When Agda can’t figure it out, we need to
supply it and write
!𝟙 {𝓤} {X} x.
This is because 𝓤 is also an implicit argument (all things declared
with the Agda keyword variable as
above
are implicit arguments). There are other,
non-positional,
ways to indicate X without having to indicate 𝓤 too. Occasionally,
people define variants of a function with different choices of
“implicitness”, as above.
𝟘It is defined like 𝟙, except that no elements are listed for it:
data 𝟘 : 𝓤₀ ̇ where
That’s the complete definition. This has a dual interpretation, mathematically as the empty set (we can actually prove that this type is a set, once we know the definition of set), and logically as the truth value false. To prove that a property of elements of the empty type holds for all elements of the empty type, we have to do nothing.
𝟘-induction : (A : 𝟘 → 𝓤 ̇ ) → (x : 𝟘) → A x 𝟘-induction A ()
When we write the pattern (), Agda checks if there is any case we
missed. If there is none, our definition is accepted. The expression
() corresponds to the mathematical phrase vacuously
true. The unique
function from 𝟘 to any type is a particular case of 𝟘-induction.
𝟘-recursion : (A : 𝓤 ̇ ) → 𝟘 → A 𝟘-recursion A a = 𝟘-induction (λ _ → A) a
We will use the following categorical notation for 𝟘-recursion:
!𝟘 : (A : 𝓤 ̇ ) → 𝟘 → A !𝟘 = 𝟘-recursion
We give the two names is-empty and ¬ to the same function now:
is-empty : 𝓤 ̇ → 𝓤 ̇ is-empty X = X → 𝟘 ¬ : 𝓤 ̇ → 𝓤 ̇ ¬ X = X → 𝟘
This says that a type is empty precisely when we have a function to
the empty type. Assuming univalence,
once we have defined the identity type former
_＝_, we will be able to prove that
(is-empty X) ＝ (X ≃ 𝟘), where X ≃ 𝟘 is the type of bijections, or
equivalences, from X to
𝟘. We will also be able to prove things like (2 + 2 ＝ 5) ＝ 𝟘 and
(2 + 2 ＝ 4) ＝ 𝟙.
This is for numbers. If we define types 𝟚 = 𝟙 + 𝟙 and 𝟜 = 𝟚 +
𝟚 with two and four elements respectively, where we are anticipating
the definition of _+_ for types, then we
will instead have that 𝟚 + 𝟚 ＝ 𝟜 is a type with 4! elements, which
is the number of permutations
of a set with four elements, rather than a truth value 𝟘 or 𝟙, as
a consequence of the univalence axiom. That is, we will have (𝟚 + 𝟚 ＝
𝟜) ≃ (𝟜 + 𝟜 + 𝟜 + 𝟜 + 𝟜 + 𝟜), so that the type identity 𝟚 + 𝟚 ＝ 𝟜
holds in many more ways than the
numerical equation 2 + 2 ＝ 4.
The above is possible only because universes are genuine types and
hence their elements (that is, types) have identity types themselves,
so that writing X ＝ Y for types X and Y (inhabiting the same
universe) is allowed.
When we view 𝟘 as false, we can read the definition of
the negation ¬ X as saying that “X implies false”. With univalence
we will be able to show that “(false → true) ＝ true”, which amounts
to (𝟘 → 𝟙) ＝ 𝟙, which in turn says that there is precisely one function
𝟘 → 𝟙, namely the vacuous function.
ℕ of natural numbersThe definition is similar but not quite the same as the one via Peano Axioms.
We stipulate an element zero : ℕ and a successor function succ : ℕ → ℕ,
and then define induction. Once we have defined the identity type former _＝_, we
will prove the other peano axioms.
data ℕ : 𝓤₀ ̇ where zero : ℕ succ : ℕ → ℕ
In general, declarations with data are inductive definitions. To write the number 5, we have to write
succ (succ (succ (succ (succ zero))))
We can use the following Agda
built-in
to be able to just write 5 as a shorthand:
{-# BUILTIN NATURAL ℕ #-}
Apart from this notational effect, the above declaration doesn’t play any role in the Agda development of these lecture notes.
In the following, the type family A can be seen as playing the role
of a property of elements of ℕ, except that it doesn’t need to be
necessarily
subsingleton valued. When it
is, the type of the function gives the familiar principle of
mathematical
induction for
natural numbers, whereas, in general, its definition says how to
compute with induction.
ℕ-induction : (A : ℕ → 𝓤 ̇ ) → A 0 → ((n : ℕ) → A n → A (succ n)) → (n : ℕ) → A n ℕ-induction A a₀ f = h where h : (n : ℕ) → A n h 0 = a₀ h (succ n) = f n (h n)
The definition of ℕ-induction is itself by induction. It says that given a point
a₀ : A 0
and a function
f : (n : ℕ) → A n → A (succ n),
in order to calculate an element of A n for a given n : ℕ, we just calculate h n, that is,
f n (f (n-1) (f (n-2) (... (f 0 a₀)...))).
So the principle of induction is a construction that given a base
case a₀ : A 0, an induction step f : (n : ℕ) → A n → A (succ n)
and a number n, says how to get an element of the type A n by
primitive
recursion.
Notice also that ℕ-induction is the dependently typed version of
primitive recursion, where the non-dependently typed version is
ℕ-recursion : (X : 𝓤 ̇ ) → X → (ℕ → X → X) → ℕ → X ℕ-recursion X = ℕ-induction (λ _ → X)
The following special case occurs often (and is related to the fact that ℕ is the initial algebra of the functor 𝟙 + (-)):
ℕ-iteration : (X : 𝓤 ̇ ) → X → (X → X) → ℕ → X ℕ-iteration X x f = ℕ-recursion X x (λ _ x → f x)
Agda checks that any recursive definition as above is well founded, with recursive invocations with structurally smaller arguments only. If it isn’t, the definition is not accepted.
In official Martin-Löf type theories, we have to use the ℕ-induction
functional to define everything else with the natural numbers. But Agda
allows us to define functions by structural recursion, like we defined
ℕ-induction.
We now define addition and multiplication for the sake of illustration.
We first do it in Peano style. We will create a local module so that the
definitions are not globally visible, as we want to have the symbols
+ and × free for type operations of MLTT to be defined soon. The
things in the module are indented and are visible outside the module
only if we open the module or if we write them as
e.g. Arithmetic._+_ in the following example.
module Arithmetic where _+_ _×_ : ℕ → ℕ → ℕ x + 0 = x x + succ y = succ (x + y) x × 0 = 0 x × succ y = x + x × y infixl 20 _+_ infixl 21 _×_
The above “fixity” declarations allow us to indicate the precedences
(multiplication has higher precedence than addition) and their
associativity (here we take left-associativity as the convention, so that
e.g. x+y+z parses as (x+y)+z).
Equivalent definitions use ℕ-induction on the second argument y, via
ℕ-iteration:
module Arithmetic' where _+_ _×_ : ℕ → ℕ → ℕ x + y = h y where h : ℕ → ℕ h = ℕ-iteration ℕ x succ x × y = h y where h : ℕ → ℕ h = ℕ-iteration ℕ 0 (x +_) infixl 20 _+_ infixl 21 _×_
Here the expression “x +_” stands for the function ℕ → ℕ that adds
x to its argument. So to multiply x by y, we apply y times the
function “x +_” to 0.
As another example, we define the less-than-or-equal relation by nested induction, on the first argument and then the second, but we use pattern matching for the sake of readability.
Exercise. Write it using
ℕ-induction, recursion or iteration, as appropriate.
module ℕ-order where _≤_ _≥_ : ℕ → ℕ → 𝓤₀ ̇ 0 ≤ y = 𝟙 succ x ≤ 0 = 𝟘 succ x ≤ succ y = x ≤ y x ≥ y = y ≤ x infix 10 _≤_ infix 10 _≥_
Exercise. After learning Σ
and _＝_ explained below, prove that
x ≤ yif and only ifΣ z ꞉ ℕ , x + z ＝ y.
Later, after learning univalence prove that in this case this implies
(x ≤ y) ＝ Σ z ꞉ ℕ , x + z ＝ y.
That bi-implication can be turned into equality only holds for types that are subsingletons (and this is called propositional extensionality).
If we are doing applied mathematics and want to actually compute, we can define a type for binary notation for the sake of efficiency, and of course people have done that. Here we are not concerned with efficiency but only with understanding how to codify mathematics in (univalent) type theory and in Agda.
_+_We now define the disjoint sum of two types X and Y. The elements of
the type
X + Y
are stipulated to be of the forms
inl xandinr y
with x : X and y : Y. If X : 𝓤 and Y : 𝓥, we stipulate that
X + Y : 𝓤 ⊔ 𝓥, where
𝓤 ⊔ 𝓥
is the least upper bound of the two universes 𝓤 and
𝓥. In Agda we can define this as follows.
data _+_ {𝓤 𝓥} (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) : 𝓤 ⊔ 𝓥 ̇ where inl : X → X + Y inr : Y → X + Y
To prove that a property A of the sum holds for all z : X + Y, it is enough to
prove that A (inl x) holds for all x : X and that A (inr y) holds for
all y : Y. This amounts to definition by cases:
+-induction : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (A : X + Y → 𝓦 ̇ ) → ((x : X) → A (inl x)) → ((y : Y) → A (inr y)) → (z : X + Y) → A z +-induction A f g (inl x) = f x +-induction A f g (inr y) = g y +-recursion : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : 𝓦 ̇ } → (X → A) → (Y → A) → X + Y → A +-recursion {𝓤} {𝓥} {𝓦} {X} {Y} {A} = +-induction (λ _ → A)
When the types A and B are understood as mathematical statements,
the type A + B is understood as the statement “A or B”, because
to prove “A or B” we have to prove one of A and B. When A and
B are simultaneously possible, we have two proofs, but sometimes we
want to deliberately ignore which one we get, when we want to get a
truth value rather than a possibly more general type, and in this case
we use the truncation ∥ A + B ∥.
But also _+_ is used to construct mathematical objects. For example,
we can define a two-point type:
𝟚 : 𝓤₀ ̇ 𝟚 = 𝟙 + 𝟙
We can name the left and right points as follows, using patterns, so that they can be used in pattern matching:
pattern ₀ = inl ⋆ pattern ₁ = inr ⋆
We can define induction on 𝟚 directly by pattern matching:
𝟚-induction : (A : 𝟚 → 𝓤 ̇ ) → A ₀ → A ₁ → (n : 𝟚) → A n 𝟚-induction A a₀ a₁ ₀ = a₀ 𝟚-induction A a₀ a₁ ₁ = a₁
Or we can prove it by induction on _+_ and 𝟙:
𝟚-induction' : (A : 𝟚 → 𝓤 ̇ ) → A ₀ → A ₁ → (n : 𝟚) → A n 𝟚-induction' A a₀ a₁ = +-induction A (𝟙-induction (λ (x : 𝟙) → A (inl x)) a₀) (𝟙-induction (λ (y : 𝟙) → A (inr y)) a₁)
Σ typesGiven universes 𝓤 and 𝓥, a type
X : 𝓤
and a type family
Y : X → 𝓥,
we want to construct its sum, which is a type whose elements are of the form
(x , y)
with x : X and y : Y x. This sum type will live in the least
upper bound
𝓤 ⊔ 𝓥
of the universes 𝓤 and 𝓥. We will write this sum
Σ Y,
with X, as well as the universes, implicit. Often Agda, and people,
can figure out what the unwritten type X is, from the definition of Y. But
sometimes there may be either lack of enough information, or of
enough concentration power by people, or of sufficiently powerful inference
algorithms in the implementation of Agda. In such cases we can write
Σ λ(x : X) → Y x,
because Y = λ (x : X) → Y x by a so-called η-rule. However, we will
define syntax to be able to write this in more familiar form as
Σ x ꞉ X , Y x.
In MLTT we can write this in other
ways, for example with the
indexing x : X written as a subscript of Σ or under it.
Or it may be that the name Y is not defined, and we work with a
nameless family defined on the fly, as in the exercise proposed above:
Σ z ꞉ ℕ , x + z ＝ y,
where Y z = (x + z ＝ y) in this case, and where we haven’t defined
the identity type former _＝_ yet.
We can construct the Σ type former as follows in Agda:
record Σ {𝓤 𝓥} {X : 𝓤 ̇ } (Y : X → 𝓥 ̇ ) : 𝓤 ⊔ 𝓥 ̇ where constructor _,_ field x : X y : Y x
This says we are defining a binary operator _,_ to construct the
elements of this type as x , y. The brackets are not needed, but we
will often write them to get the more familiar notation (x , y). The
definition says that an element of Σ Y has two fields, giving the
types for them.
Remark. Beginners may safely ignore this remark: Normally people
will call these two fields x and y something like pr₁ and pr₂,
or fst and snd, for first and second projection, rather than x
and y, and then do open Σ public and have the projections
available as functions automatically. But we will deliberately not do
that, and instead define the projections ourselves, because this is
confusing for beginners, no matter how mathematically or
computationally versed they may be, in particular because it will not
be immediately clear that the projections have the following types.
pr₁ : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } → Σ Y → X pr₁ (x , y) = x pr₂ : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } → (z : Σ Y) → Y (pr₁ z) pr₂ (x , y) = y
We now introduce syntax to be able to write Σ x ꞉ X , Y instead of
Σ λ(x ꞉ X) → Y. For this purpose, we first define a version of Σ
making the index type explicit.
-Σ : {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) (Y : X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ -Σ X Y = Σ Y syntax -Σ X (λ x → y) = Σ x ꞉ X , y
For some reason, Agda has this kind of definition backwards: the definiendum and the definiens are swapped with respect to the normal convention of writing what is defined on the left-hand side of the equality sign.
Notice also that “꞉” in the above syntax definition is not the same
as “:”, even though they may look the same. For the above notation Σ
x ꞉ A , b, the symbol “꞉” has to be typed “\:4” in the emacs Agda
mode.
To prove that A z holds for all z : Σ Y, for a given
property A, we just prove that we have A (x , y) for all x :
X and y : Y x. This is called Σ induction or Σ
elimination, or uncurry, after Haskell
Curry.
Σ-induction : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {A : Σ Y → 𝓦 ̇ } → ((x : X) (y : Y x) → A (x , y)) → ((x , y) : Σ Y) → A (x , y) Σ-induction g (x , y) = g x y
This function has an inverse:
curry : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {A : Σ Y → 𝓦 ̇ } → (((x , y) : Σ Y) → A (x , y)) → ((x : X) (y : Y x) → A (x , y)) curry f x y = f (x , y)
An important particular case of the sum type is the binary cartesian product, when the type family doesn’t depend on the indexing type:
_×_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ X × Y = Σ x ꞉ X , Y
We have seen by way of examples that the function type symbol →
represents logical implication, and that a dependent function type (x : X) → A x
represents a universal quantification. We have the
following uses of Σ.
The binary cartesian product represents conjunction “and”. If the
types A and B stand for mathematical statements, then the
mathematical statement
AandB
is codified as
A × B,
because to establish “A and B”, we have to provide a
pair (a , b) of proofs a : A and b : B.
So notice that in type theory proofs are mathematical objects, rather than meta-mathematical entities like in set theory. They are just elements of types.
The more general type
Σ x : X , A x,
if the type X stands for a mathematical object and A stands for a mathematical
statement, represents designated existence
there is a designated
x : XwithA x.
To establish this, we have to provide a specific element x : X
and a proof a : A x, together in a pair (x , a).
Later we will discuss unspecified existence
∃ x : X , A x,
which will be obtained by a sort of quotient of Σ x : X , A x,
written
∥ Σ x : X , A x ∥,
that identifies all the elements of the type Σ x : X , A x in
a single equivalence class, called its subsingleton (or truth
value or propositional) truncation.
Another reading of
Σ x : X , A x
is as
the type of
x : XwithA x,
similar to set-theoretical notation
{ x ∈ X | A x }.
But we
have to be careful because if there is more than one element
in the type A x, then x will occur more than once in this
type. More precisely, for a₀ a₁ : A x we have inhabitants
(x , a₀) and (x , a₁) of the type Σ x : X , A x.
In such situations, if we don’t want that, we have to either
ensure that the type A x has at most one element for every x :
X, or instead consider the truncated type ∥ A x ∥ and write
Σ x : X , ∥ A x ∥.
An example is the image of a function f : X → Y,
which will be defined to be
Σ y : Y , ∥ Σ x : X , f x ＝ y ∥.
This is the type of y : Y for which there is an unspecified
x : X with f x ＝ y.
(For constructively minded readers, we emphasize that this
doesn’t erase the
witness x : X.)
Π typesΠ types are builtin with a different notation in Agda, as discussed
above, but we can introduce the notation Π for them, similar to that for Σ:
Π : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ Π {𝓤} {𝓥} {X} A = (x : X) → A x -Π : {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) (Y : X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ -Π X Y = Π Y syntax -Π A (λ x → b) = Π x ꞉ A , b
Notice that the function type X → Y is the particular case of the Π
type when the family A is constant with value Y.
We take the opportunity to define the identity function (in two versions with different implicit arguments) and function composition:
id : {X : 𝓤 ̇ } → X → X id x = x 𝑖𝑑 : (X : 𝓤 ̇ ) → X → X 𝑖𝑑 X = id
Usually the type of function composition _∘_ is given as simply
(Y → Z) → (X → Y) → (X → Z).
With dependent functions, we can give it a more general type:
_∘_ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : Y → 𝓦 ̇ } → ((y : Y) → Z y) → (f : X → Y) → (x : X) → Z (f x) g ∘ f = λ x → g (f x)
Notice that this type for the composition function can be read as a mathematical
statement: If Z y holds for all y : Y, then for any given f : X →
Y we have that Z (f x) holds for all x : X. And the non-dependent
one is just the transitivity of implication.
The following functions are sometimes useful when we are using implicit arguments, in order to recover them explicitly without having to list them as given arguments:
domain : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ̇ domain {𝓤} {𝓥} {X} {Y} f = X codomain : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓥 ̇ codomain {𝓤} {𝓥} {X} {Y} f = Y type-of : {X : 𝓤 ̇ } → X → 𝓤 ̇ type-of {𝓤} {X} x = X
Id, also written _＝_We now introduce the central type constructor of MLTT from the point of view of univalent mathematics. In Agda we can define Martin-Löf’s identity type as follows:
data Id {𝓤} (X : 𝓤 ̇ ) : X → X → 𝓤 ̇ where refl : (x : X) → Id X x x
Intuitively, the above definition would say that the only element
of the type Id X x x is something called refl x (for
reflexivity). But, as we shall see in a moment, this intuition turns
out to be incorrect.
Notice a crucial difference with the previous definitions using data
or induction: In the previous cases, we defined types, namely 𝟘,
𝟙, ℕ, or a type depending on type parameters, namely _+_, with 𝓤
and 𝓥 fixed,
_+_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇
But here we are defining a type family indexed by the elements of
a given type, rather than a new type from old types. Given a type X
in a universe 𝓤, we define a function
Id X : X → X → 𝓤
by some mysterious sort of induction. It is this that prevents us from
being able to prove that the only element of the type Id X x x would
be refl x, or that the type Id X x y would have at most one
element no matter what y : X is.
There is however, one interesting, and crucial, thing we can
prove, namely that for any fixed
element x : X, the type
Σ y ꞉ Y , Id X x y
is always a singleton.
We will use the following alternative notation for the identity type
former Id, where the symbol “_” in the right-hand side of the
definition indicates that we ask Agda to infer which type we are
talking about (which is X, but this name is not available in the
scope of the defining equation of the type former _＝_):
_＝_ : {X : 𝓤 ̇ } → X → X → 𝓤 ̇ x ＝ y = Id _ x y
Notational remark. The symbol “＝” for the identity type is full width equal sign because the normal Ascii symbol “=” is reserved in Agda. To set-up emacs to type our symbol for the identity type, please follow these instructions.
Another intuition for this type family _＝_ : X → X → 𝓤 is that it
gives the least reflexive relation on the type X, as suggested by
Martin-Löf’s induction principle J discussed below.
Whereas we can make the intuition that x ＝ x has precisely one
element good by postulating a certain K
axiom due to
Thomas Streicher, which comes with Agda by default but we have
disabled above, we cannot
prove that refl x is the only element of x ＝ x for an arbitrary
type X. This non-provability result was established by Hofmann and
Streicher, by giving a
model of type theory in which types are interpreted as
1-groupoids. This is in
spirit similar to the non-provability of the parallel
postulate in
Euclidean geometry, which also considers models, which in turn are
interesting in their own right.
However, for the elements of some types, such as the type ℕ of
natural numbers, it is possible to
prove that any identity type x ＝ y
has at most one element. Such types are called sets in univalent
mathematics.
If instead of the axiom K we adopt Voevodsky’s
univalence axiom, we get specific
examples of objects x and y such that
the type x ＝ y has multiple elements, within the type theory. It
follows that the identity type x ＝ y is fairly under-specified in
general, in that we can’t prove or disprove that it has at most one
element.
There are two opposing ways to resolve the ambiguity or
under-specification of the identity types: (1) We can consider the K
axiom, which postulates that all types are sets, or (2) we can
consider the univalence axiom, arriving at univalent mathematics,
which gives rise to types that are more general than sets, the
n-groupoids and ∞-groupoids. In fact, the univalence axiom will
say, in particular, that for some types X and elements x y : X, the
identity type x ＝ y does have more than one element.
A possible way to understand the element refl x of the type x ＝ x
is as the “generic identification” between the point x and itself,
but which is by no means necessarily the only identitification in
univalent foundations. It is generic in the sense that to explain what
happens with all identifications p : x ＝ y between any two points
x and y of a type X, it suffices to explain what happens with
the identification refl x : x ＝ x for all points x : X. This is
what the induction principle for identity given by Martin-Löf says,
which he called J (we could have called it ＝-induction, but we
prefer to honour MLTT tradition):
𝕁 : (X : 𝓤 ̇ ) (A : (x y : X) → x ＝ y → 𝓥 ̇ ) → ((x : X) → A x x (refl x)) → (x y : X) (p : x ＝ y) → A x y p 𝕁 X A f x x (refl x) = f x
This is related to the Yoneda
Lemma in category theory,
for readers familiar with the subject, which says that certain natural
transformations are uniquely determined by their action on the
identity arrows, even if they are defined for all arrows. Similarly
here, 𝕁 is uniquely determined by its action on reflexive
identifications, but is defined for all identifications between any
two points, not just reflexivities.
In summary, Martin-Löf’s identity type is given by the data
Id,refl,𝕁,𝕁.However, we will not always use this induction principle, because we
can instead work with the instances we need by pattern matching on
refl (which is just what we did to define the principle itself) and
there is a theorem by Jesper
Cockx that says that
with the Agda option without-K, pattern matching on refl can
define/prove precisely what 𝕁 can.
Exercise. Define
ℍ : {X : 𝓤 ̇ } (x : X) (B : (y : X) → x ＝ y → 𝓥 ̇ ) → B x (refl x) → (y : X) (p : x ＝ y) → B y p ℍ x B b x (refl x) = b
Then we can define 𝕁 from ℍ as follows:
𝕁' : (X : 𝓤 ̇ ) (A : (x y : X) → x ＝ y → 𝓥 ̇ ) → ((x : X) → A x x (refl x)) → (x y : X) (p : x ＝ y) → A x y p 𝕁' X A f x = ℍ x (A x) (f x) 𝕁s-agreement : (X : 𝓤 ̇ ) (A : (x y : X) → x ＝ y → 𝓥 ̇ ) (f : (x : X) → A x x (refl x)) (x y : X) (p : x ＝ y) → 𝕁 X A f x y p ＝ 𝕁' X A f x y p 𝕁s-agreement X A f x x (refl x) = refl (f x)
Similarly define ℍ' from 𝕁 without using pattern matching on refl
and show that it coincides with ℍ (possibly using pattern matching
on refl). This is harder.
With this, we have concluded the rendering of our spartan MLTT in Agda notation. Before embarking on the development of univalent mathematics within our spartan MLTT, we pause to discuss some basic examples of mathematics in Martin-Löf type theory.
Transport along an identification.
transport : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} → x ＝ y → A x → A y transport A (refl x) = 𝑖𝑑 (A x)
We can equivalently define transport using 𝕁 as follows:
transport𝕁 : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} → x ＝ y → A x → A y transport𝕁 {𝓤} {𝓥} {X} A {x} {y} = 𝕁 X (λ x y _ → A x → A y) (λ x → 𝑖𝑑 (A x)) x y
In the same way ℕ-recursion can be seen as the non-dependent special
case of ℕ-induction, the following transport function can be seen as
the non-dependent special case of the ＝-induction principle ℍ with
some of the arguments permuted and made implicit:
nondep-ℍ : {X : 𝓤 ̇ } (x : X) (A : X → 𝓥 ̇ ) → A x → (y : X) → x ＝ y → A y nondep-ℍ x A = ℍ x (λ y _ → A y) transportℍ : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} → x ＝ y → A x → A y transportℍ A {x} {y} p a = nondep-ℍ x A a y p
All the above transports coincide:
transports-agreement : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} (p : x ＝ y) → (transportℍ A p ＝ transport A p) × (transport𝕁 A p ＝ transport A p) transports-agreement A (refl x) = refl (transport A (refl x)) , refl (transport A (refl x))
The following is for use when we want to recover implicit arguments without mentioning them.
lhs : {X : 𝓤 ̇ } {x y : X} → x ＝ y → X lhs {𝓤} {X} {x} {y} p = x rhs : {X : 𝓤 ̇ } {x y : X} → x ＝ y → X rhs {𝓤} {X} {x} {y} p = y
Composition of identifications.
Given two identifications p : x ＝ y and q : y ＝ z, we can compose them
to get an identification p ∙ q : x ＝ z. This can also be seen as
transitivity of equality. Because the type of composition doesn’t
mention p and q, we can use the non-dependent version of ＝-induction.
_∙_ : {X : 𝓤 ̇ } {x y z : X} → x ＝ y → y ＝ z → x ＝ z p ∙ q = transport (lhs p ＝_) q p
Here we are considering the family A t = (x ＝ t), and using the
identification q : y ＝ z to transport A y to A z, that is x ＝
y to x ＝ z.
Exercise. Can you define an alternative version that uses p to
transport. Do the two versions give equal results?
When writing p ∙ q, we lose information on the lhs and the rhs of the
identifications p : x ＝ y and q : y ＝ z, which makes some definitions hard to read. We now
introduce notation to be able to write e.g.
x ＝⟨ p ⟩
y ＝⟨ q ⟩
z ∎
as a synonym of the expression p ∙ q with some of the implicit arguments of _∙_ made
explicit. We have one ternary mixfix operator _＝⟨_⟩_ and one unary
postfix operator _∎.
_＝⟨_⟩_ : {X : 𝓤 ̇ } (x : X) {y z : X} → x ＝ y → y ＝ z → x ＝ z x ＝⟨ p ⟩ q = p ∙ q _∎ : {X : 𝓤 ̇ } (x : X) → x ＝ x x ∎ = refl x
Inversion of identifications. Given an identification, we get an identification in the opposite direction:
_⁻¹ : {X : 𝓤 ̇ } → {x y : X} → x ＝ y → y ＝ x p ⁻¹ = transport (_＝ lhs p) p (refl (lhs p))
We can define an alternative of identification composition with this:
_∙'_ : {X : 𝓤 ̇ } {x y z : X} → x ＝ y → y ＝ z → x ＝ z p ∙' q = transport (_＝ rhs q) (p ⁻¹) q
This agrees with the previous one:
∙agreement : {X : 𝓤 ̇ } {x y z : X} (p : x ＝ y) (q : y ＝ z) → p ∙' q ＝ p ∙ q ∙agreement (refl x) (refl x) = refl (refl x)
But refl y is a definitional neutral element for one of them on the right and for the other one on the left,
p ∙ refl y = p,refl y ∙' q = q,which can be checked as follows
rdnel : {X : 𝓤 ̇ } {x y : X} (p : x ＝ y) → p ∙ refl y ＝ p rdnel p = refl p rdner : {X : 𝓤 ̇ } {y z : X} (q : y ＝ z) → refl y ∙' q ＝ q rdner q = refl q
Exercise. The identification refl y is neutral on both sides of
each of the two operations _∙_ and _∙'_, although not
definitionally. This has to be proved by induction on
identifications, as in ∙-agreement.
Application of a function to an identification.
Given an identification p : x ＝ x' we get an identification
ap f p : f x ＝ f x' for any f : X → Y:
ap : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) {x x' : X} → x ＝ x' → f x ＝ f x' ap f {x} {x'} p = transport (λ - → f x ＝ f -) p (refl (f x))
Here the symbol “-”, which is not to be confused with the symbol
“_”, is a variable. We will adopt the convention in these notes of
using this variable name “-” to make clear which part of an
expression we are replacing with transport.
Notice that we have so far used the recursion principle transport
only. To reason about transport, _∙_, _⁻¹ and ap, we will
need to use the full induction
principle 𝕁 (or equivalently pattern matching on refl).
Pointwise equality of functions. We will work with pointwise equality of functions, defined as follows, which, using univalence, will be equivalent to equality of functions.
_∼_ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → Π A → Π A → 𝓤 ⊔ 𝓥 ̇ f ∼ g = ∀ x → f x ＝ g x
The symbol ∀ is a built-in notation for Π . We could equivalently
write the definiens as
(x : _) → f x ＝ g x,
or, with our domain notation
(x : domain f) → f x ＝ g x.
We first introduce notation for double and triple negation to avoid the use of brackets.
¬¬ ¬¬¬ : 𝓤 ̇ → 𝓤 ̇ ¬¬ A = ¬(¬ A) ¬¬¬ A = ¬(¬¬ A)
To prove that A → ¬¬ A, that is, A → ((A → 𝟘) → 𝟘), we start with
a hypothetical element a : A and a hypothetical function u : A → 𝟘
and the goal is to get an element of 𝟘. All we need to do is to
apply the function u to a. This gives double-negation
introduction:
dni : (A : 𝓤 ̇ ) → A → ¬¬ A dni A a u = u a
Mathematically, this says that if we have a point of A (we say that
A is pointed) then A is nonempty. There is no general procedure to
implement the converse, that is, from a function (A → 𝟘) → 𝟘 to get
a point of A. For truth
values A, we can assume
this as an axiom if we wish, because it is equivalent to the
principle excluded middle. For arbitrary types A,
this would be a form of global
choice for type
theory. However, global choice is inconsistent with univalence [HoTT
book, Theorem 3.2.2], because
there is no way to choose an element of every non-empty type in a way
that is invariant under automorphisms. But the axiom of
choice is consistent with univalent type
theory, as stated in the introduction.
In the proof of the following, we are given hypothetical
functions f : A → B and v : B → 𝟘, and a hypothetical element a :
A, and our goal is to get an element of 𝟘. But this is easy,
because f a : B and hence v (f a) : 𝟘.
contrapositive : {A : 𝓤 ̇ } {B : 𝓥 ̇ } → (A → B) → (¬ B → ¬ A) contrapositive f v a = v (f a)
We have given a logical name to this function. Mathematically, this
says that if we have a function A → B and B is empty, then A
must be empty, too. The proof is by assuming that A would have an
inhabitant a, to get a contradiction, namely that B would have an
inhabitant, too, showing that there can’t be any such inhabitant a
of A after all. See
Bauer
for a general discussion.
And from this we get that three negations imply one:
tno : (A : 𝓤 ̇ ) → ¬¬¬ A → ¬ A tno A = contrapositive (dni A)
Hence, using dni once again, we get that ¬¬¬ A if and only if ¬
A. It is entertaining to see how Brouwer formulated and proved this
fact in his Cambridge Lectures on
Intuitionism:
Theorem. Absurdity of absurdity of absurdity is equivalent to absurdity.
Proof. Firstly, since implication of the assertion 𝑦 by the assertion 𝑥 implies implication of absurdity of 𝑥 by absurdity of 𝑦, the implication of absurdity of absurdity by truth (which is an established fact) implies the implication of absurdity of truth, that is to say of absurdity, by absurdity of absurdity of absurdity. Secondly, since truth of an assertion implies absurdity of its absurdity, in particular truth of absurdity implies absurdity of absurdity of absurdity.
If we define logical equivalence by
_⇔_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ X ⇔ Y = (X → Y) × (Y → X) lr-implication : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X ⇔ Y) → (X → Y) lr-implication = pr₁ rl-implication : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X ⇔ Y) → (Y → X) rl-implication = pr₂
then we can render Brouwer’s argument in Agda as follows, where the
“established fact” is dni:
absurdity³-is-absurdity : {A : 𝓤 ̇ } → ¬¬¬ A ⇔ ¬ A absurdity³-is-absurdity {𝓤} {A} = firstly , secondly where firstly : ¬¬¬ A → ¬ A firstly = contrapositive (dni A) secondly : ¬ A → ¬¬¬ A secondly = dni (¬ A)
But of course Brouwer, as is well known, was averse to formalism, and hence wouldn’t approve of such a sacrilege.
We now define a symbol for the negation of equality.
_≠_ : {X : 𝓤 ̇ } → X → X → 𝓤 ̇ x ≠ y = ¬(x ＝ y)
In the following proof, we have u : x ＝ y → 𝟘 and need to define a
function y ＝ x → 𝟘. So all we need to do is to compose the function
that inverts identifications with u:
≠-sym : {X : 𝓤 ̇ } {x y : X} → x ≠ y → y ≠ x ≠-sym {𝓤} {X} {x} {y} u = λ (p : y ＝ x) → u (p ⁻¹)
To show that the type 𝟙 is not equal to the type 𝟘, we use that
transport id gives 𝟙 ＝ 𝟘 → id 𝟙 → id 𝟘 where id is the identity
function of the universe 𝓤₀. More
generally, we have the following conversion of type identifications
into functions:
Id→Fun : {X Y : 𝓤 ̇ } → X ＝ Y → X → Y Id→Fun {𝓤} = transport (𝑖𝑑 (𝓤 ̇ ))
Here the identity function is that of the universe 𝓤 where the types
X and Y live. An equivalent definition is the following, where
this time the identity function is that of the type X:
Id→Fun' : {X Y : 𝓤 ̇ } → X ＝ Y → X → Y Id→Fun' (refl X) = 𝑖𝑑 X Id→Funs-agree : {X Y : 𝓤 ̇ } (p : X ＝ Y) → Id→Fun p ＝ Id→Fun' p Id→Funs-agree (refl X) = refl (𝑖𝑑 X)
So if we have a hypothetical identification p : 𝟙 ＝ 𝟘, then we get a
function 𝟙 → 𝟘. We apply this function to ⋆ : 𝟙 to conclude the
proof.
𝟙-is-not-𝟘 : 𝟙 ≠ 𝟘 𝟙-is-not-𝟘 p = Id→Fun p ⋆
To show that the elements ₁ and ₀ of the two-point type 𝟚 are
not equal, we reduce to the above case. We start with a hypothetical
identification p : ₁ ＝ ₀.
₁-is-not-₀ : ₁ ≠ ₀ ₁-is-not-₀ p = 𝟙-is-not-𝟘 q where f : 𝟚 → 𝓤₀ ̇ f ₀ = 𝟘 f ₁ = 𝟙 q : 𝟙 ＝ 𝟘 q = ap f p
Remark. Agda allows us to use a pattern () to get the following
quick proof. However, this method of proof doesn’t belong to the
realm of MLTT. Hence we will use the pattern () only in the above
definition of 𝟘-induction and
nowhere else in these notes.
₁-is-not-₀[not-an-MLTT-proof] : ¬(₁ ＝ ₀) ₁-is-not-₀[not-an-MLTT-proof] ()
Perhaps the following is sufficiently self-explanatory given the above:
decidable : 𝓤 ̇ → 𝓤 ̇ decidable A = A + ¬ A has-decidable-equality : 𝓤 ̇ → 𝓤 ̇ has-decidable-equality X = (x y : X) → decidable (x ＝ y) 𝟚-has-decidable-equality : has-decidable-equality 𝟚 𝟚-has-decidable-equality ₀ ₀ = inl (refl ₀) 𝟚-has-decidable-equality ₀ ₁ = inr (≠-sym ₁-is-not-₀) 𝟚-has-decidable-equality ₁ ₀ = inr ₁-is-not-₀ 𝟚-has-decidable-equality ₁ ₁ = inl (refl ₁)
So we consider four cases. In the first and the last, we have equal
things and so we give an answer in the left-hand side of the sum. In
the middle two, we give an answer in the right-hand side, where we need
functions ₀ ＝ ₁ → 𝟘 and ₁ ＝ ₀ → 𝟘, which we can take to be ≠-sym
₁-is-not-₀ and ₁-is-not-₀ respectively.
The following is more interesting. We consider the two possible cases
for n. The first one assumes a hypothetical function f : ₀ ＝ ₀ →
𝟘, from which we get f (refl ₀) : 𝟘, and then, using !𝟘, we get
an element of any type we like, which we choose to be ₀ ＝ ₁, and we
are done. Of course, we will never be able to use the function
not-zero-is-one with such outrageous arguments. The other case n =
₁ doesn’t need to use the hypothesis f : ₁ ＝ ₀ → 𝟘, because the
desired conclusion holds right away, as it is ₁ ＝ ₁, which is proved
by refl ₁. But notice that there is nothing wrong with the
hypothesis f : ₁ ＝ ₀ → 𝟘. For example, we can use not-zero-is-one
taking n to be ₀ and f to be ₁-is-not-₀, so that the
hypotheses can be fulfilled in the second equation.
not-zero-is-one : (n : 𝟚) → n ≠ ₀ → n ＝ ₁ not-zero-is-one ₀ f = !𝟘 (₀ ＝ ₁) (f (refl ₀)) not-zero-is-one ₁ f = refl ₁
The following generalizes ₁-is-not-₀, both in its statement and its
proof (so we could have formulated it first and then used it to deduce
₁-is-not-₀):
inl-inr-disjoint-images : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {x : X} {y : Y} → inl x ≠ inr y inl-inr-disjoint-images {𝓤} {𝓥} {X} {Y} p = 𝟙-is-not-𝟘 q where f : X + Y → 𝓤₀ ̇ f (inl x) = 𝟙 f (inr y) = 𝟘 q : 𝟙 ＝ 𝟘 q = ap f p
If P or Q holds and Q fails, then P holds:
right-fails-gives-left-holds : {P : 𝓤 ̇ } {Q : 𝓥 ̇ } → P + Q → ¬ Q → P right-fails-gives-left-holds (inl p) u = p right-fails-gives-left-holds (inr q) u = !𝟘 _ (u q)
We illustrate the above constructs of MLTT to formulate this conjecture.
module twin-primes where open Arithmetic renaming (_×_ to _*_ ; _+_ to _∔_) open ℕ-order is-prime : ℕ → 𝓤₀ ̇ is-prime n = (n ≥ 2) × ((x y : ℕ) → x * y ＝ n → (x ＝ 1) + (x ＝ n)) twin-prime-conjecture : 𝓤₀ ̇ twin-prime-conjecture = (n : ℕ) → Σ p ꞉ ℕ , (p ≥ n) × is-prime p × is-prime (p ∔ 2)
Thus, not only can we write down definitions, constructions, theorems and proofs, but also conjectures. They are just definitions of types. Likewise, the univalence axiom, to be formulated in due course, is a type.
We first prove the remaining Peano axioms.
positive-not-zero : (x : ℕ) → succ x ≠ 0 positive-not-zero x p = 𝟙-is-not-𝟘 (g p) where f : ℕ → 𝓤₀ ̇ f 0 = 𝟘 f (succ x) = 𝟙 g : succ x ＝ 0 → 𝟙 ＝ 𝟘 g = ap f
To show that the successor function is left cancellable, we can use the following predecessor function.
pred : ℕ → ℕ pred 0 = 0 pred (succ n) = n succ-lc : {x y : ℕ} → succ x ＝ succ y → x ＝ y succ-lc = ap pred
With this we have proved all the Peano axioms.
Without assuming the principle of excluded middle, we can prove that
ℕ has decidable equality:
ℕ-has-decidable-equality : has-decidable-equality ℕ ℕ-has-decidable-equality 0 0 = inl (refl 0) ℕ-has-decidable-equality 0 (succ y) = inr (≠-sym (positive-not-zero y)) ℕ-has-decidable-equality (succ x) 0 = inr (positive-not-zero x) ℕ-has-decidable-equality (succ x) (succ y) = f (ℕ-has-decidable-equality x y) where f : decidable (x ＝ y) → decidable (succ x ＝ succ y) f (inl p) = inl (ap succ p) f (inr k) = inr (λ (s : succ x ＝ succ y) → k (succ-lc s))
Exercise. Students should do this kind of thing at least once in
their academic life: rewrite the above proof of the decidability of
equality of ℕ to use the ℕ-induction principle instead of pattern
matching and recursion, to understand by themselves that this can be
done.
We now move to basic arithmetic, and we use a module for that.
module basic-arithmetic-and-order where open ℕ-order public open Arithmetic renaming (_+_ to _∔_) hiding (_×_)
We can show that addition is associative as follows, by induction on
z, where IH stands for “induction hypothesis”:
+-assoc : (x y z : ℕ) → (x ∔ y) ∔ z ＝ x ∔ (y ∔ z) +-assoc x y 0 = (x ∔ y) ∔ 0 ＝⟨ refl _ ⟩ x ∔ (y ∔ 0) ∎ +-assoc x y (succ z) = (x ∔ y) ∔ succ z ＝⟨ refl _ ⟩ succ ((x ∔ y) ∔ z) ＝⟨ ap succ IH ⟩ succ (x ∔ (y ∔ z)) ＝⟨ refl _ ⟩ x ∔ (y ∔ succ z) ∎ where IH : (x ∔ y) ∔ z ＝ x ∔ (y ∔ z) IH = +-assoc x y z
Notice that the proofs refl _ should be read as “by definition” or
“by construction”. They are not necessary, because Agda knows the
definitions and silently expands them when necessary, but we are
writing them here for the sake of clarity.
Here is
the version with the silent expansion of definitions, for the sake of
illustration (the author of these notes can write, but not read it the
absence of the above verbose version):
+-assoc' : (x y z : ℕ) → (x ∔ y) ∔ z ＝ x ∔ (y ∔ z) +-assoc' x y zero = refl _ +-assoc' x y (succ z) = ap succ (+-assoc' x y z)
We defined addition by induction on the second argument. Next we show that the base case and induction step of a definition by induction on the first argument hold (but of course not definitionally). We do this by induction on the second argument.
+-base-on-first : (x : ℕ) → 0 ∔ x ＝ x +-base-on-first 0 = refl 0 +-base-on-first (succ x) = 0 ∔ succ x ＝⟨ refl _ ⟩ succ (0 ∔ x) ＝⟨ ap succ IH ⟩ succ x ∎ where IH : 0 ∔ x ＝ x IH = +-base-on-first x +-step-on-first : (x y : ℕ) → succ x ∔ y ＝ succ (x ∔ y) +-step-on-first x zero = refl (succ x) +-step-on-first x (succ y) = succ x ∔ succ y ＝⟨ refl _ ⟩ succ (succ x ∔ y) ＝⟨ ap succ IH ⟩ succ (x ∔ succ y) ∎ where IH : succ x ∔ y ＝ succ (x ∔ y) IH = +-step-on-first x y
Using this, the commutativity of addition can be proved by induction on the first argument.
+-comm : (x y : ℕ) → x ∔ y ＝ y ∔ x +-comm 0 y = 0 ∔ y ＝⟨ +-base-on-first y ⟩ y ＝⟨ refl _ ⟩ y ∔ 0 ∎ +-comm (succ x) y = succ x ∔ y ＝⟨ +-step-on-first x y ⟩ succ(x ∔ y) ＝⟨ ap succ IH ⟩ succ(y ∔ x) ＝⟨ refl _ ⟩ y ∔ succ x ∎ where IH : x ∔ y ＝ y ∔ x IH = +-comm x y
We now show that addition is cancellable in its left argument, by induction on the left argument:
+-lc : (x y z : ℕ) → x ∔ y ＝ x ∔ z → y ＝ z +-lc 0 y z p = y ＝⟨ (+-base-on-first y)⁻¹ ⟩ 0 ∔ y ＝⟨ p ⟩ 0 ∔ z ＝⟨ +-base-on-first z ⟩ z ∎ +-lc (succ x) y z p = IH where q = succ (x ∔ y) ＝⟨ (+-step-on-first x y)⁻¹ ⟩ succ x ∔ y ＝⟨ p ⟩ succ x ∔ z ＝⟨ +-step-on-first x z ⟩ succ (x ∔ z) ∎ IH : y ＝ z IH = +-lc x y z (succ-lc q)
Now we solve part of an exercise given above, namely that (x ≤ y) ⇔ Σ z ꞉ ℕ , x + z ＝ y.
First we name the alternative definition of ≤:
_≼_ : ℕ → ℕ → 𝓤₀ ̇ x ≼ y = Σ z ꞉ ℕ , x ∔ z ＝ y
Next we show that the two relations ≤ and ≼ imply each other.
In both cases, we proceed by induction on both arguments.
≤-gives-≼ : (x y : ℕ) → x ≤ y → x ≼ y ≤-gives-≼ 0 0 l = 0 , refl 0 ≤-gives-≼ 0 (succ y) l = succ y , +-base-on-first (succ y) ≤-gives-≼ (succ x) 0 l = !𝟘 (succ x ≼ zero) l ≤-gives-≼ (succ x) (succ y) l = γ where IH : x ≼ y IH = ≤-gives-≼ x y l z : ℕ z = pr₁ IH p : x ∔ z ＝ y p = pr₂ IH γ : succ x ≼ succ y γ = z , (succ x ∔ z ＝⟨ +-step-on-first x z ⟩ succ (x ∔ z) ＝⟨ ap succ p ⟩ succ y ∎) ≼-gives-≤ : (x y : ℕ) → x ≼ y → x ≤ y ≼-gives-≤ 0 0 (z , p) = ⋆ ≼-gives-≤ 0 (succ y) (z , p) = ⋆ ≼-gives-≤ (succ x) 0 (z , p) = positive-not-zero (x ∔ z) q where q = succ (x ∔ z) ＝⟨ (+-step-on-first x z)⁻¹ ⟩ succ x ∔ z ＝⟨ p ⟩ zero ∎ ≼-gives-≤ (succ x) (succ y) (z , p) = IH where q = succ (x ∔ z) ＝⟨ (+-step-on-first x z)⁻¹ ⟩ succ x ∔ z ＝⟨ p ⟩ succ y ∎ IH : x ≤ y IH = ≼-gives-≤ x y (z , succ-lc q)
Later we will show that
(x ≤ y) ＝ Σ z ꞉ ℕ , x + z ＝ y, using univalence.
We now develop some generally useful material regarding the order ≤
on natural numbers. First, it is reflexive, transitive and antisymmetric:
≤-refl : (n : ℕ) → n ≤ n ≤-refl zero = ⋆ ≤-refl (succ n) = ≤-refl n ≤-trans : (l m n : ℕ) → l ≤ m → m ≤ n → l ≤ n ≤-trans zero m n p q = ⋆ ≤-trans (succ l) zero n p q = !𝟘 (succ l ≤ n) p ≤-trans (succ l) (succ m) zero p q = q ≤-trans (succ l) (succ m) (succ n) p q = ≤-trans l m n p q ≤-anti : (m n : ℕ) → m ≤ n → n ≤ m → m ＝ n ≤-anti zero zero p q = refl zero ≤-anti zero (succ n) p q = !𝟘 (zero ＝ succ n) q ≤-anti (succ m) zero p q = !𝟘 (succ m ＝ zero) p ≤-anti (succ m) (succ n) p q = ap succ (≤-anti m n p q) ≤-succ : (n : ℕ) → n ≤ succ n ≤-succ zero = ⋆ ≤-succ (succ n) = ≤-succ n zero-minimal : (n : ℕ) → zero ≤ n zero-minimal n = ⋆ unique-minimal : (n : ℕ) → n ≤ zero → n ＝ zero unique-minimal zero p = refl zero unique-minimal (succ n) p = !𝟘 (succ n ＝ zero) p ≤-split : (m n : ℕ) → m ≤ succ n → (m ≤ n) + (m ＝ succ n) ≤-split zero n l = inl l ≤-split (succ m) zero l = inr (ap succ (unique-minimal m l)) ≤-split (succ m) (succ n) l = +-recursion inl (inr ∘ ap succ) (≤-split m n l) _<_ : ℕ → ℕ → 𝓤₀ ̇ x < y = succ x ≤ y infix 10 _<_ not-<-gives-≥ : (m n : ℕ) → ¬(n < m) → m ≤ n not-<-gives-≥ zero n u = zero-minimal n not-<-gives-≥ (succ m) zero u = dni (zero < succ m) (zero-minimal m) u not-<-gives-≥ (succ m) (succ n) u = not-<-gives-≥ m n u bounded-∀-next : (A : ℕ → 𝓤 ̇ ) (k : ℕ) → A k → ((n : ℕ) → n < k → A n) → (n : ℕ) → n < succ k → A n bounded-∀-next A k a φ n l = +-recursion f g s where s : (n < k) + (succ n ＝ succ k) s = ≤-split (succ n) k l f : n < k → A n f = φ n g : succ n ＝ succ k → A n g p = transport A ((succ-lc p)⁻¹) a
The type of roots of a function:
root : (ℕ → ℕ) → 𝓤₀ ̇ root f = Σ n ꞉ ℕ , f n ＝ 0 _has-no-root<_ : (ℕ → ℕ) → ℕ → 𝓤₀ ̇ f has-no-root< k = (n : ℕ) → n < k → f n ≠ 0 is-minimal-root : (ℕ → ℕ) → ℕ → 𝓤₀ ̇ is-minimal-root f m = (f m ＝ 0) × (f has-no-root< m) at-most-one-minimal-root : (f : ℕ → ℕ) (m n : ℕ) → is-minimal-root f m → is-minimal-root f n → m ＝ n at-most-one-minimal-root f m n (p , φ) (q , ψ) = c m n a b where a : ¬(m < n) a u = ψ m u p b : ¬(n < m) b v = φ n v q c : (m n : ℕ) → ¬(m < n) → ¬(n < m) → m ＝ n c m n u v = ≤-anti m n (not-<-gives-≥ m n v) (not-<-gives-≥ n m u)
The type of minimal roots of a function:
minimal-root : (ℕ → ℕ) → 𝓤₀ ̇ minimal-root f = Σ m ꞉ ℕ , is-minimal-root f m minimal-root-is-root : ∀ f → minimal-root f → root f minimal-root-is-root f (m , p , _) = m , p bounded-ℕ-search : ∀ k f → (minimal-root f) + (f has-no-root< k) bounded-ℕ-search zero f = inr (λ n → !𝟘 (f n ≠ 0)) bounded-ℕ-search (succ k) f = +-recursion φ γ (bounded-ℕ-search k f) where A : ℕ → (ℕ → ℕ) → 𝓤₀ ̇ A k f = (minimal-root f) + (f has-no-root< k) φ : minimal-root f → A (succ k) f φ = inl γ : f has-no-root< k → A (succ k) f γ u = +-recursion γ₀ γ₁ (ℕ-has-decidable-equality (f k) 0) where γ₀ : f k ＝ 0 → A (succ k) f γ₀ p = inl (k , p , u) γ₁ : f k ≠ 0 → A (succ k) f γ₁ v = inr (bounded-∀-next (λ n → f n ≠ 0) k v u)
Given any root, we can find a minimal root.
root-gives-minimal-root : ∀ f → root f → minimal-root f root-gives-minimal-root f (n , p) = γ where g : ¬(f has-no-root< (succ n)) g φ = φ n (≤-refl n) p γ : minimal-root f γ = right-fails-gives-left-holds (bounded-ℕ-search (succ n) f) g
But, as discussed above, rather than postulating univalence and truncation, we will use them as explicit assumptions each time they are needed.
We emphasize that there are univalent type theories in which univalence and existence of truncations are theorems, for example cubical type theory, which has a version available in Agda, called cubical Agda.
Voevodsky defined a notion of contractible type, which we refer to
here as singleton type. We say that a type is a singleton if there
is a designated c : X that is identified with each x : X.
is-singleton : 𝓤 ̇ → 𝓤 ̇ is-singleton X = Σ c ꞉ X , ((x : X) → c ＝ x)
Such an element c is sometimes referred to as a center of
contraction of X, in connection with homotopy theory.
is-center : (X : 𝓤 ̇ ) → X → 𝓤 ̇ is-center X c = (x : X) → c ＝ x
The canonical singleton type is 𝟙:
𝟙-is-singleton : is-singleton 𝟙 𝟙-is-singleton = ⋆ , 𝟙-induction (λ x → ⋆ ＝ x) (refl ⋆)
Once we have defined the notion of type
equivalence, we will have
that a type is a singleton if and only if it is equivalent to 𝟙.
When working with singleton types, it will be convenient to have distinguished names for the two projections:
center : (X : 𝓤 ̇ ) → is-singleton X → X center X (c , φ) = c centrality : (X : 𝓤 ̇ ) (i : is-singleton X) (x : X) → center X i ＝ x centrality X (c , φ) = φ
A type is a subsingleton if it has at most one element, that is, any two of its elements are equal, or identified.
is-subsingleton : 𝓤 ̇ → 𝓤 ̇ is-subsingleton X = (x y : X) → x ＝ y 𝟘-is-subsingleton : is-subsingleton 𝟘 𝟘-is-subsingleton x y = !𝟘 (x ＝ y) x singletons-are-subsingletons : (X : 𝓤 ̇ ) → is-singleton X → is-subsingleton X singletons-are-subsingletons X (c , φ) x y = x ＝⟨ (φ x)⁻¹ ⟩ c ＝⟨ φ y ⟩ y ∎ 𝟙-is-subsingleton : is-subsingleton 𝟙 𝟙-is-subsingleton = singletons-are-subsingletons 𝟙 𝟙-is-singleton pointed-subsingletons-are-singletons : (X : 𝓤 ̇ ) → X → is-subsingleton X → is-singleton X pointed-subsingletons-are-singletons X x s = (x , s x) singleton-iff-pointed-and-subsingleton : {X : 𝓤 ̇ } → is-singleton X ⇔ (X × is-subsingleton X) singleton-iff-pointed-and-subsingleton {𝓤} {X} = (a , b) where a : is-singleton X → X × is-subsingleton X a s = center X s , singletons-are-subsingletons X s b : X × is-subsingleton X → is-singleton X b (x , t) = pointed-subsingletons-are-singletons X x t
The terminology stands for subtype
of a singleton and is
justified
by the fact that a type X is a subsingleton according to the above
definition if and only if the map X → 𝟙 is an
embedding, if and only if X is
embedded into a singleton.
Under univalent excluded
middle, the empty type 𝟘 and the singleton
type 𝟙 are the only subsingletons, up to equivalence, or up to
identity if we further assume univalence.
Subsingletons are also called propositions or truth values:
is-prop is-truth-value : 𝓤 ̇ → 𝓤 ̇ is-prop = is-subsingleton is-truth-value = is-subsingleton
The terminology is-subsingleton is more mathematical and avoids the
clash with the slogan propositions as
types,
which is based on the interpretation of mathematical statements as
arbitrary types, rather than just subsingletons. In these notes we
prefer the terminology subsingleton, but we occasionally use the
terminologies proposition and truth value. When we wish to
emphasize this notion of proposition adopted in univalent mathematics,
as opposed to “propositions as (arbitrary) types”, we may say
univalent proposition.
In some formal systems, for example set theory based on first-order logic, one can tell that something is a proposition by looking at its syntactical form, e.g. “there are infinitely many prime numbers” is a proposition, by construction, in such a theory.
In univalent mathematics, however, propositions are mathematical,
rather than meta-mathematical, objects, and the fact that a type turns
out to be a proposition requires mathematical proof. Moreover, such a
proof may be subtle and not immediate and require significant
preparation. An example is
the proof that the univalence axiom is a proposition, which relies on
the fact that univalence implies
function extensionality, which in turn
implies that the
statement that a map is an equivalence is a proposition. Another
non-trivial example, which again relies on univalence or at least
function extensionality, is the proof that the statement that a type
X is a proposition is itself a
proposition,
which can be written as is-prop (is-prop X).
Singletons and subsingletons are also called -2-groupoids and -1-groupoids respectively.
A type is defined to be a set if there is at most one way for any two of its elements to be equal:
is-set : 𝓤 ̇ → 𝓤 ̇ is-set X = (x y : X) → is-subsingleton (x ＝ y)
At this point, with the definition of these notions, we are entering the realm of univalent mathematics, but not yet needing the univalence axiom.
As mentioned above, under excluded middle, the only two subsingletons,
up to equivalence, are 𝟘 and 𝟙. In fact, excluded middle in
univalent mathematics says precisely that.
EM EM' : ∀ 𝓤 → 𝓤 ⁺ ̇ EM 𝓤 = (X : 𝓤 ̇ ) → is-subsingleton X → X + ¬ X EM' 𝓤 = (X : 𝓤 ̇ ) → is-subsingleton X → is-singleton X + is-empty X
Notice that the above two definitions don’t assert excluded middle, but instead say what excluded middle is (like when we said what the twin-prime conjecture is), in two logically equivalent ways:
EM-gives-EM' : EM 𝓤 → EM' 𝓤 EM-gives-EM' em X s = γ (em X s) where γ : X + ¬ X → is-singleton X + is-empty X γ (inl x) = inl (pointed-subsingletons-are-singletons X x s) γ (inr ν) = inr ν EM'-gives-EM : EM' 𝓤 → EM 𝓤 EM'-gives-EM em' X s = γ (em' X s) where γ : is-singleton X + is-empty X → X + ¬ X γ (inl i) = inl (center X i) γ (inr e) = inr e
We will not assume or deny excluded middle, which is an independent statement in our spartan univalent type theory - it can’t be proved or disproved, just as the parallel postulate in Euclidean geometry can’t be proved or disproved. We will deliberately keep it undecided, adopting a neutral approach to the constructive vs. non-constructive dichotomy. We will however prove a couple of consequences of excluded middle in discussions of foundational issues such as size and existence of subsingleton truncations. We will also prove that excluded middle is a consequence of the axiom of choice.
It should be emphasized that the potential failure of excluded middle doesn’t say that there may be mysterious subsingletons that fail to be singletons and also fail to be empty. No such things occur in mathematical nature:
no-unicorns : ¬(Σ X ꞉ 𝓤 ̇ , is-subsingleton X × ¬(is-singleton X) × ¬(is-empty X)) no-unicorns (X , i , f , g) = c where e : is-empty X e x = f (pointed-subsingletons-are-singletons X x i) c : 𝟘 c = g e
Given this, what does the potential failure of excluded middle mean?
That there is no general way, provided by our spartan univalent type
theory, to determine which of the two cases is-singleton X and
is-empty X applies for a given subsingleton X. This kind of
phenomenon should be familiar even in classical, non-constructive
mathematics: although we are entitled to believe that the Goldbach
conjecture either holds or fails, we still don’t know which one is the
case, despite efforts by the sharpest mathematical minds. A
hypothetical element of the type EM would, in particular, be able to
solve the Goldbach conjecture. There is nothing wrong or contradictory
with assuming the existence of such a magic blackbox (EM stands for
“there Exists such a Magic box”). There is only loss of
generality
and of the implicit algorithmic character of our spartan base type
theory, both of which most mathematicians will be perfectly happy to live
with.
In these notes we don’t advocate any particular philosophy for or against excluded middle and other non-constructive principles. We confine ourselves to discussing mathematical facts. Axioms that can be assumed consistently but reduce generality and/or break the implicit computational character of our base type theory are discussed at various parts of these lecture notes, and are summarized at the end.
Exercise. We also have that it is impossible for is-singleton X +
is-empty X to fail for a given subsingleton X, which amounts to
saying that
¬¬(is-singleton X + is-empty X)
always holds.
Occasionaly one hears people asserting that this says that the double negation of excluded middle holds. But this is incorrect. The above statement, when written in full, is
(X : 𝓤 ̇ ) → is-subsingleton X → ¬¬(is-singleton X + is-empty X).
This is a theorem, which is quite different from the double negation of excluded middle, which is not a theorem and is
¬¬((X : 𝓤 ̇ ) → is-subsingleton X → is-singleton X + is-empty X).
Just as excluded middle, this is an independent statement.
Exercise. Continued from the previous exercise. Also for any type
R replacing the empty type, there is a function ((X + (X → R)) → R)
→ R, so that the kind of phenomenon illustrated in the previous
exercise has little to do with the emptiness of the empty type.
A magma is a set equipped with a binary operation subject to no laws
[Bourbaki]. We can define the type of magmas in a universe 𝓤 as follows:
module magmas where Magma : (𝓤 : Universe) → 𝓤 ⁺ ̇ Magma 𝓤 = Σ X ꞉ 𝓤 ̇ , is-set X × (X → X → X)
The type Magma 𝓤 collects all magmas in a universe 𝓤, and lives in
the successor universe 𝓤 ⁺. Thus, this doesn’t define what a magma is as
a property. It defines the type of magmas. A magma is an element of
this type, that is, a triple (X , i , _·_) with X : 𝓤 and i :
is-set X and _·_ : X → X → X.
Given a magma M = (X , i , _·_) we denote by ⟨ M ⟩ its underlying
set X and by magma-operation M its multiplication _·_:
⟨_⟩ : Magma 𝓤 → 𝓤 ̇ ⟨ X , i , _·_ ⟩ = X magma-is-set : (M : Magma 𝓤) → is-set ⟨ M ⟩ magma-is-set (X , i , _·_) = i magma-operation : (M : Magma 𝓤) → ⟨ M ⟩ → ⟨ M ⟩ → ⟨ M ⟩ magma-operation (X , i , _·_) = _·_
The following syntax declaration
allows us to write x ·⟨ M ⟩ y as an abbreviation of magma-operation M x y:
syntax magma-operation M x y = x ·⟨ M ⟩ y
The point is that this time we need
such a mechanism because in order to be able to mention arbitrary x
and y, we first need to know their types, which is ⟨ M ⟩ and hence
M has to occur before x and y in the definition of
magma-operation. The syntax declaration circumvents this.
A function of the underlying sets of two magmas is a called a homomorphism when it commutes with the magma operations:
is-magma-hom : (M N : Magma 𝓤) → (⟨ M ⟩ → ⟨ N ⟩) → 𝓤 ̇ is-magma-hom M N f = (x y : ⟨ M ⟩) → f (x ·⟨ M ⟩ y) ＝ f x ·⟨ N ⟩ f y id-is-magma-hom : (M : Magma 𝓤) → is-magma-hom M M (𝑖𝑑 ⟨ M ⟩) id-is-magma-hom M = λ (x y : ⟨ M ⟩) → refl (x ·⟨ M ⟩ y) is-magma-iso : (M N : Magma 𝓤) → (⟨ M ⟩ → ⟨ N ⟩) → 𝓤 ̇ is-magma-iso M N f = is-magma-hom M N f × (Σ g ꞉ (⟨ N ⟩ → ⟨ M ⟩), is-magma-hom N M g × (g ∘ f ∼ 𝑖𝑑 ⟨ M ⟩) × (f ∘ g ∼ 𝑖𝑑 ⟨ N ⟩)) id-is-magma-iso : (M : Magma 𝓤) → is-magma-iso M M (𝑖𝑑 ⟨ M ⟩) id-is-magma-iso M = id-is-magma-hom M , 𝑖𝑑 ⟨ M ⟩ , id-is-magma-hom M , refl , refl
Any identification of magmas gives rise to a magma isomorphism by transport:
Id→iso : {M N : Magma 𝓤} → M ＝ N → ⟨ M ⟩ → ⟨ N ⟩ Id→iso p = transport ⟨_⟩ p Id→iso-is-iso : {M N : Magma 𝓤} (p : M ＝ N) → is-magma-iso M N (Id→iso p) Id→iso-is-iso (refl M) = id-is-magma-iso M
The isomorphisms can be collected in a type:
_≅ₘ_ : Magma 𝓤 → Magma 𝓤 → 𝓤 ̇ M ≅ₘ N = Σ f ꞉ (⟨ M ⟩ → ⟨ N ⟩), is-magma-iso M N f
The following function will be a bijection in the presence of univalence, so that the identifications of magmas are in one-to-one correspondence with the magma isomorphisms:
magma-Id→iso : {M N : Magma 𝓤} → M ＝ N → M ≅ₘ N magma-Id→iso p = (Id→iso p , Id→iso-is-iso p)
If we omit the sethood condition in the definition of the type of
magmas, we get the type of what we could call ∞-magmas (then the
type of magmas could be called 0-Magma).
∞-Magma : (𝓤 : Universe) → 𝓤 ⁺ ̇ ∞-Magma 𝓤 = Σ X ꞉ 𝓤 ̇ , (X → X → X)
A monoid is a set equipped with an associative binary operation and with a two-sided neutral element, and so we get the type of monoids as follows.
We first define the three laws:
module monoids where left-neutral : {X : 𝓤 ̇ } → X → (X → X → X) → 𝓤 ̇ left-neutral e _·_ = ∀ x → e · x ＝ x right-neutral : {X : 𝓤 ̇ } → X → (X → X → X) → 𝓤 ̇ right-neutral e _·_ = ∀ x → x · e ＝ x associative : {X : 𝓤 ̇ } → (X → X → X) → 𝓤 ̇ associative _·_ = ∀ x y z → (x · y) · z ＝ x · (y · z)
Then a monoid is a set equipped with such e and _·_ satisfying these
three laws:
Monoid : (𝓤 : Universe) → 𝓤 ⁺ ̇ Monoid 𝓤 = Σ X ꞉ 𝓤 ̇ , is-set X × (Σ · ꞉ (X → X → X) , (Σ e ꞉ X , (left-neutral e ·) × (right-neutral e ·) × (associative ·)))
Remark. People are more likely to use
records
in Agda rather than iterated Σs as above (recall that we defined
Σ using a record). This is fine, because records amount to iterated
Σ types (recall that also _×_ is a Σ type, by
definition). Here, however, we are being deliberately spartan. Once we
have defined our Agda notation for MLTT, we want to stick to
it. This is for teaching purposes (of MLTT, encoded in Agda, not of
Agda itself in its full glory).
We could drop the is-set X condition, but then we wouldn’t get
∞-monoids in any reasonable sense. We would instead get “wild
∞-monoids” or “incoherent ∞-monoids”. The reason is that in
monoids (with sets as carriers) the neutrality and associativity
equations can hold in at most one way, by definition of set. But if we
drop the sethood requirement, then the equations can hold in multiple
ways. And then one is forced to consider equations between the
identifications (all the way up in the ∞-case), which is
what “coherence data”
means. The wildness or incoherence amounts to the absence of such
data.
Similarly to the situation with magmas, identifications of monoids are in bijection with monoid isomorphisms, assuming univalence. For this to be the case, it is absolutely necessary that the carrier of a monoid is a set rather than an arbitrary type, for otherwise the monoid equations can hold in many possible ways, and we would need to consider a notion of monoid isomorphism that in addition to preserving the neutral element and the multiplication, preserves the associativity identifications.
Exercise. Define the type of groups (with sets as carriers).
Exercise. Write down the various types of categories defined in the HoTT book in Agda.
Exercise. Try to define a type of topological spaces.
We can view a type X as a sort of category with hom-types rather than
hom-sets, with the identifications between points as the arrows.
We have that refl provides a neutral element for composition of
identifications:
refl-left : {X : 𝓤 ̇ } {x y : X} {p : x ＝ y} → refl x ∙ p ＝ p refl-left {𝓤} {X} {x} {x} {refl x} = refl (refl x) refl-right : {X : 𝓤 ̇ } {x y : X} {p : x ＝ y} → p ∙ refl y ＝ p refl-right {𝓤} {X} {x} {y} {p} = refl p
And composition is associative:
∙assoc : {X : 𝓤 ̇ } {x y z t : X} (p : x ＝ y) (q : y ＝ z) (r : z ＝ t) → (p ∙ q) ∙ r ＝ p ∙ (q ∙ r) ∙assoc p q (refl z) = refl (p ∙ q)
If we wanted to prove the above without pattern matching, this time we
would need the dependent version 𝕁 of induction on _＝_.
Exercise. Try to do this with 𝕁 and with ℍ.
But all arrows, the identifications, are invertible:
⁻¹-left∙ : {X : 𝓤 ̇ } {x y : X} (p : x ＝ y) → p ⁻¹ ∙ p ＝ refl y ⁻¹-left∙ (refl x) = refl (refl x) ⁻¹-right∙ : {X : 𝓤 ̇ } {x y : X} (p : x ＝ y) → p ∙ p ⁻¹ ＝ refl x ⁻¹-right∙ (refl x) = refl (refl x)
A category in which all arrows are invertible is called a groupoid. The above is the basis for the Hofmann–Streicher groupoid model of type theory.
But we actually get higher groupoids, because given identifications
p q : x ＝ y
we can consider the identity type p ＝ q, and given
u v : p ＝ q
we can consider the type u ＝ v, and so on.
See [van den Berg and Garner] and
[Lumsdaine].
For some types, such as the natural numbers, we can
prove that this process trivializes
after the first step, because the type x ＝ y has at most one
element. Such types are the sets as defined above.
Voevodsky defined the notion of hlevel to measure how long it takes for the process to trivialize.
Here are some more constructions with identifications:
⁻¹-involutive : {X : 𝓤 ̇ } {x y : X} (p : x ＝ y) → (p ⁻¹)⁻¹ ＝ p ⁻¹-involutive (refl x) = refl (refl x)
The application operation on identifications is functorial, in the sense that it preserves the neutral element and commutes with composition:
ap-refl : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (x : X) → ap f (refl x) ＝ refl (f x) ap-refl f x = refl (refl (f x)) ap-∙ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) {x y z : X} (p : x ＝ y) (q : y ＝ z) → ap f (p ∙ q) ＝ ap f p ∙ ap f q ap-∙ f p (refl y) = refl (ap f p)
Notice that we also have
ap⁻¹ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) {x y : X} (p : x ＝ y) → (ap f p)⁻¹ ＝ ap f (p ⁻¹) ap⁻¹ f (refl x) = refl (refl (f x))
The above functions ap-refl and ap-∙ constitute functoriality in
the second argument. We also have functoriality in the first argument,
in the following sense:
ap-id : {X : 𝓤 ̇ } {x y : X} (p : x ＝ y) → ap id p ＝ p ap-id (refl x) = refl (refl x) ap-∘ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) (g : Y → Z) {x y : X} (p : x ＝ y) → ap (g ∘ f) p ＝ (ap g ∘ ap f) p ap-∘ f g (refl x) = refl (refl (g (f x)))
Transport is also functorial with respect to identification composition and function composition. By construction, it maps the neutral element to the identity function. The apparent contravariance takes place because we have defined function composition in the usual order, but identification composition in the diagramatic order (as is customary in each case).
transport∙ : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y z : X} (p : x ＝ y) (q : y ＝ z) → transport A (p ∙ q) ＝ transport A q ∘ transport A p transport∙ A p (refl y) = refl (transport A p)
Functions of a type into a universe can be considered as generalized
presheaves, which we could perhaps call ∞-presheaves. Their morphisms
are natural transformations:
Nat : {X : 𝓤 ̇ } → (X → 𝓥 ̇ ) → (X → 𝓦 ̇ ) → 𝓤 ⊔ 𝓥 ⊔ 𝓦 ̇ Nat A B = (x : domain A) → A x → B x
We don’t need to specify the naturality condition, because it is automatic:
Nats-are-natural : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (B : X → 𝓦 ̇ ) (τ : Nat A B) → {x y : X} (p : x ＝ y) → τ y ∘ transport A p ＝ transport B p ∘ τ x Nats-are-natural A B τ (refl x) = refl (τ x)
We will use the following constructions a number of times:
NatΣ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → Nat A B → Σ A → Σ B NatΣ τ (x , a) = (x , τ x a) transport-ap : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (A : Y → 𝓦 ̇ ) (f : X → Y) {x x' : X} (p : x ＝ x') (a : A (f x)) → transport (A ∘ f) p a ＝ transport A (ap f p) a transport-ap A f (refl x) a = refl a
If we have an identification p : A ＝ B of two types A and B, and
elements a : A and b : B, we cannot ask directly whether a ＝ b,
because although the types are identified by p, they are not
necessarily the same, in the sense of definitional equality. This is
not merely a syntactical restriction of our formal system, but instead
a fundamental fact that reflects the philosophy of univalent
mathematics. For instance, consider the type
data Color : 𝓤₀ ̇ where Black White : Color
With univalence, we will have that Color ＝ 𝟚 where 𝟚 is the
two-point type 𝟙 + 𝟙 with elements ₀ and
₁. But there will be two identifications p₀ p₁ : Color ＝ 𝟚, one
that identifies Black with ₀ and White with ₁, and another one
that identifies Black with ₁ and White with ₀. There is no
preferred coding of binary colors as bits. And, precisely because of
that, even if univalence does give inhabitants of the type Color ＝
𝟚, it doesn’t make sense to ask whether Black ＝ ₀ holds without
specifying one of the possible inhabitants p₀ and p₁.
What we will have is that the functions transport id p₀ and
transport id p₁ are the two possible bijections Color → 𝟚 that
identify colors with bits. So, it is not enough to have Color ＝ 𝟚 to
be able to compare a color c : Color with a bit b : 𝟚. We need to
specify which identification p : Color ＝ 𝟚 we want to consider for
the comparison. The same considerations
apply when we consider identifications p : 𝟚 ＝ 𝟚.
So the meaningful comparison in the more general situation is
transport id p a ＝ b
for a specific
p : A ＝ B,
where id is the identity function of the universe where the types A
and B live, and hence
transport id : A ＝ B → (A → B)
is the function that transforms identifications into functions, which has already occurred above.
More generally, we want to consider the situation in which we replace
the identity function id of the universe where A and B live by
an arbitrary type family, which is what we do now.
If we have a type
X : 𝓤 ̇,
and a type family
A : X → 𝓥 ̇
and points
x y : X
and an identification
p : x ＝ y,
then we get the identification
ap A p : A x ＝ A y.
However, if we have
a : A x,
b : A y,
we again cannot write down the identity type
.a ＝ b
This is again a non-sensical mathematical statement, because the types
A x and A y are not the same, but only identified, and in general
there can be many identifications, not just ap A p, and so any
identification between elements of A x and A y has to be with
respect to a specific identification, as in the above particular case.
This time, the meaningful comparison, given p : x ＝ y, is
transport A p a ＝ b,
For example, this idea applies when comparing the values of a dependent function:
apd : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (f : (x : X) → A x) {x y : X} (p : x ＝ y) → transport A p (f x) ＝ f y apd f (refl x) = refl (f x)
With the above notion of dependent equality, we can characterize
equality in Σ types as follows.
to-Σ-＝ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {σ τ : Σ A} → (Σ p ꞉ pr₁ σ ＝ pr₁ τ , transport A p (pr₂ σ) ＝ pr₂ τ) → σ ＝ τ to-Σ-＝ (refl x , refl a) = refl (x , a) from-Σ-＝ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {σ τ : Σ A} → σ ＝ τ → Σ p ꞉ pr₁ σ ＝ pr₁ τ , transport A p (pr₂ σ) ＝ pr₂ τ from-Σ-＝ (refl (x , a)) = (refl x , refl a)
For the sake of readability, the above two definitions can be equivalently written as follows.
to-Σ-＝-again : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {(x , a) (y , b) : Σ A} → Σ p ꞉ x ＝ y , transport A p a ＝ b → (x , a) ＝ (y , b) to-Σ-＝-again (refl x , refl a) = refl (x , a) from-Σ-＝-again : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {(x , a) (y , b) : Σ A} → (x , a) ＝ (y , b) → Σ p ꞉ x ＝ y , transport A p a ＝ b from-Σ-＝-again (refl (x , a)) = (refl x , refl a)
The above gives the logical equivalence
(σ ＝ τ) ⇔ (Σ p ꞉ pr₁ σ ＝ pr₁ τ , transport A p (pr₂ σ) ＝ pr₂ τ).
But this is a very weak statement when the left- and right-hand identity types have multiple elements, which is precisely the point of univalent mathematics.
What we want is the lhs and the rhs to be isomorphic, or more precisely, equivalent in the sense of Voevodsky.
Once we have defined this notion _≃_ of type equivalence, this
characterization will become an equivalence
(σ ＝ τ) ≃ (Σ p ꞉ pr₁ σ ＝ pr₁ τ , transport A p pr₂ σ ＝ pr₂ τ).
But even this is not sufficiently precise, because in general there are multiple equivalences between two types. For example, there are precisely two equivalences
𝟙 + 𝟙 ≃ 𝟙 + 𝟙,
namely the identity function and the function that flips left and
right. What we want to say is that a specific map is an
equivalence. In our case, it is the function from-Σ-＝ defined
above.
Voevodsky came up with a definition of a type “f is an equivalence”
which is always a subsingleton: a given function f can be an
equivalence in at most one way. In other words, being an equivalence
is property of f, rather than data for f.
The following special case of to-Σ-＝ is often useful:
to-Σ-＝' : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {x : X} {a a' : A x} → a ＝ a' → Id (Σ A) (x , a) (x , a') to-Σ-＝' {𝓤} {𝓥} {X} {A} {x} = ap (λ - → (x , -))
We take the opportunity to establish more equations for transport and to define a dependent version of transport:
transport-× : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (B : X → 𝓦 ̇ ) {x y : X} (p : x ＝ y) {(a , b) : A x × B x} → transport (λ x → A x × B x) p (a , b) ＝ (transport A p a , transport B p b) transport-× A B (refl _) = refl _ transportd : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (B : (x : X) → A x → 𝓦 ̇ ) {x : X} (a : A x) {y : X} (p : x ＝ y) → B x a → B y (transport A p a) transportd A B a (refl x) = id transport-Σ : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (B : (x : X) → A x → 𝓦 ̇ ) {x : X} (y : X) (p : x ＝ y) (a : A x) {b : B x a} → transport (λ x → Σ y ꞉ A x , B x y) p (a , b) ＝ transport A p a , transportd A B a p b transport-Σ A B {x} x (refl x) a {b} = refl (a , b)
Voevodsky’s hlevels 0,1,2,3,... are shifted by 2 with respect to
the n-groupoid numbering convention, and correspond to -2-groupoids
(singletons), -1-groupoids (subsingletons), 0-groupoids (sets),…
The hlevel relation is defined by induction on ℕ, with the
induction step working with the identity types of the elements of the
type in question:
_is-of-hlevel_ : 𝓤 ̇ → ℕ → 𝓤 ̇ X is-of-hlevel 0 = is-singleton X X is-of-hlevel (succ n) = (x x' : X) → ((x ＝ x') is-of-hlevel n)
It is often convenient in practice to have equivalent formulations of
the types of hlevel 1 (as subsingletons) and 2 (as sets), which we will
develop soon.
To characterize sets as the types of hlevel 2, we first need to show that subsingletons are sets, and this is not easy. We use an argument due to Hedberg. This argument also shows that Voevodsky’s hlevels are upper closed.
We choose to present an alternative formulation of Hedberg’s Theorem, but we stress that the method of proof is essentially the same.
We first define a notion of constant map:
wconstant : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ wconstant f = (x x' : domain f) → f x ＝ f x'
The prefix “w” officially stands for “weakly”. Perhaps
incoherently constant or wildly constant would be better
terminologies, with coherence understood in the ∞-categorical
sense. We prefer to stick to wildly rather than weakly, and luckily
both start with the letter “w”.
We first define the type of constant endomaps of a given type:
wconstant-endomap : 𝓤 ̇ → 𝓤 ̇ wconstant-endomap X = Σ f ꞉ (X → X), wconstant f wcmap : (X : 𝓤 ̇ ) → wconstant-endomap X → (X → X) wcmap X (f , w) = f wcmap-constancy : (X : 𝓤 ̇ ) (c : wconstant-endomap X) → wconstant (wcmap X c) wcmap-constancy X (f , w) = w
The point is that a type is a set if and only if its identity types
all have designated wconstant endomaps:
Hedberg : {X : 𝓤 ̇ } (x : X) → ((y : X) → wconstant-endomap (x ＝ y)) → (y : X) → is-subsingleton (x ＝ y) Hedberg {𝓤} {X} x c y p q = p ＝⟨ a y p ⟩ (f x (refl x))⁻¹ ∙ f y p ＝⟨ ap (λ - → (f x (refl x))⁻¹ ∙ -) (κ y p q) ⟩ (f x (refl x))⁻¹ ∙ f y q ＝⟨ (a y q)⁻¹ ⟩ q ∎ where f : (y : X) → x ＝ y → x ＝ y f y = wcmap (x ＝ y) (c y) κ : (y : X) (p q : x ＝ y) → f y p ＝ f y q κ y = wcmap-constancy (x ＝ y) (c y) a : (y : X) (p : x ＝ y) → p ＝ (f x (refl x))⁻¹ ∙ f y p a x (refl x) = (⁻¹-left∙ (f x (refl x)))⁻¹
We consider types whose identity types all have designated constant endomaps:
wconstant-＝-endomaps : 𝓤 ̇ → 𝓤 ̇ wconstant-＝-endomaps X = (x y : X) → wconstant-endomap (x ＝ y)
The following is immediate from the definitions.
sets-have-wconstant-＝-endomaps : (X : 𝓤 ̇ ) → is-set X → wconstant-＝-endomaps X sets-have-wconstant-＝-endomaps X s x y = (f , κ) where f : x ＝ y → x ＝ y f p = p κ : (p q : x ＝ y) → f p ＝ f q κ p q = s x y p q
And the converse is the content of Hedberg’s Theorem.
types-with-wconstant-＝-endomaps-are-sets : (X : 𝓤 ̇ ) → wconstant-＝-endomaps X → is-set X types-with-wconstant-＝-endomaps-are-sets X c x = Hedberg x (λ y → wcmap (x ＝ y) (c x y) , wcmap-constancy (x ＝ y) (c x y))
In the following definition of the auxiliary function f, we ignore
the argument p, using the fact that X is a subsingleton instead,
to get a wconstant function:
subsingletons-have-wconstant-＝-endomaps : (X : 𝓤 ̇ ) → is-subsingleton X → wconstant-＝-endomaps X subsingletons-have-wconstant-＝-endomaps X s x y = (f , κ) where f : x ＝ y → x ＝ y f p = s x y κ : (p q : x ＝ y) → f p ＝ f q κ p q = refl (s x y)
And the corollary is that (sub)singleton types are sets.
subsingletons-are-sets : (X : 𝓤 ̇ ) → is-subsingleton X → is-set X subsingletons-are-sets X s = types-with-wconstant-＝-endomaps-are-sets X (subsingletons-have-wconstant-＝-endomaps X s) singletons-are-sets : (X : 𝓤 ̇ ) → is-singleton X → is-set X singletons-are-sets X = subsingletons-are-sets X ∘ singletons-are-subsingletons X
In particular, the types 𝟘 and 𝟙 are sets.
𝟘-is-set : is-set 𝟘 𝟘-is-set = subsingletons-are-sets 𝟘 𝟘-is-subsingleton 𝟙-is-set : is-set 𝟙 𝟙-is-set = subsingletons-are-sets 𝟙 𝟙-is-subsingleton
Then with the above we get our desired characterization of the types of
hlevel 1 as an immediate consequence:
subsingletons-are-of-hlevel-1 : (X : 𝓤 ̇ ) → is-subsingleton X → X is-of-hlevel 1 subsingletons-are-of-hlevel-1 X = g where g : ((x y : X) → x ＝ y) → (x y : X) → is-singleton (x ＝ y) g t x y = t x y , subsingletons-are-sets X t x y (t x y) types-of-hlevel-1-are-subsingletons : (X : 𝓤 ̇ ) → X is-of-hlevel 1 → is-subsingleton X types-of-hlevel-1-are-subsingletons X = f where f : ((x y : X) → is-singleton (x ＝ y)) → (x y : X) → x ＝ y f s x y = center (x ＝ y) (s x y)
This is an “if and only if” characterization, but, under univalence, it becomes an equality because the types under consideration are subsingletons.
The same comments as for the previous section apply.
sets-are-of-hlevel-2 : (X : 𝓤 ̇ ) → is-set X → X is-of-hlevel 2 sets-are-of-hlevel-2 X = g where g : ((x y : X) → is-subsingleton (x ＝ y)) → (x y : X) → (x ＝ y) is-of-hlevel 1 g t x y = subsingletons-are-of-hlevel-1 (x ＝ y) (t x y) types-of-hlevel-2-are-sets : (X : 𝓤 ̇ ) → X is-of-hlevel 2 → is-set X types-of-hlevel-2-are-sets X = f where f : ((x y : X) → (x ＝ y) is-of-hlevel 1) → (x y : X) → is-subsingleton (x ＝ y) f s x y = types-of-hlevel-1-are-subsingletons (x ＝ y) (s x y)
A singleton is a subsingleton, a subsingleton is a set, … , a type
of hlevel n is of hlevel n+1 too, …
Again we can conclude this almost immediately from the above development:
hlevel-upper : (X : 𝓤 ̇ ) (n : ℕ) → X is-of-hlevel n → X is-of-hlevel (succ n) hlevel-upper X zero = γ where γ : is-singleton X → (x y : X) → is-singleton (x ＝ y) γ h x y = p , subsingletons-are-sets X k x y p where k : is-subsingleton X k = singletons-are-subsingletons X h p : x ＝ y p = k x y hlevel-upper X (succ n) = λ h x y → hlevel-upper (x ＝ y) n (h x y)
To be consistent with the above terminology, we have to stipulate that
all types have hlevel ∞. We don’t need a definition for this
notion. But what may happen (and it does with univalence) is that
there are types which don’t have any finite hlevel. We can say that
such types then have minimal hlevel ∞.
Exercise. Formulate and prove the following. The type 𝟙 has
minimal hlevel 0.
_has-minimal-hlevel_ : 𝓤 ̇ → ℕ → 𝓤 ̇ X has-minimal-hlevel 0 = X is-of-hlevel 0 X has-minimal-hlevel (succ n) = (X is-of-hlevel (succ n)) × ¬(X is-of-hlevel n) _has-minimal-hlevel-∞ : 𝓤 ̇ → 𝓤 ̇ X has-minimal-hlevel-∞ = (n : ℕ) → ¬(X is-of-hlevel n)
The type 𝟘 has minimal hlevel 1, the type ℕ has minimal hlevel
2. The solution to the fact that ℕ has hlevel 2 is given in the
next section. More ambitiously, after
univalence is available, show that the
type of monoids has minimal hlevel 3.
ℕ and 𝟚 are setsIf a type has decidable equality we can define a wconstant
function x ＝ y → x ＝ y and hence conclude that it is a set. This
argument is due to Hedberg.
pointed-types-have-wconstant-endomap : {X : 𝓤 ̇ } → X → wconstant-endomap X pointed-types-have-wconstant-endomap x = ((λ y → x) , (λ y y' → refl x)) empty-types-have-wconstant-endomap : {X : 𝓤 ̇ } → is-empty X → wconstant-endomap X empty-types-have-wconstant-endomap e = (id , (λ x x' → !𝟘 (x ＝ x') (e x))) decidable-has-wconstant-endomap : {X : 𝓤 ̇ } → decidable X → wconstant-endomap X decidable-has-wconstant-endomap (inl x) = pointed-types-have-wconstant-endomap x decidable-has-wconstant-endomap (inr e) = empty-types-have-wconstant-endomap e hedberg-lemma : {X : 𝓤 ̇ } → has-decidable-equality X → wconstant-＝-endomaps X hedberg-lemma {𝓤} {X} d x y = decidable-has-wconstant-endomap (d x y) hedberg : {X : 𝓤 ̇ } → has-decidable-equality X → is-set X hedberg {𝓤} {X} d = types-with-wconstant-＝-endomaps-are-sets X (hedberg-lemma d) ℕ-is-set : is-set ℕ ℕ-is-set = hedberg ℕ-has-decidable-equality 𝟚-is-set : is-set 𝟚 𝟚-is-set = hedberg 𝟚-has-decidable-equality
Notice that excluded middle implies directly that all sets have decidable equality, so that in its presence a type is a set if and only if it has decidable equality.
We use retracts as a mathematical technique to transfer properties
between types. For instance, retracts of singletons are
singletons. Showing that a particular type X is a singleton may be
rather difficult to do directly by applying the definition of
singleton and the definition of the particular type, but it may be
easy to show that X is a retract of Y for a type Y that is
already known to be a singleton. In these notes, a major application
will be to get a simple proof of the known fact that invertible maps
are equivalences in the sense of Voevodsky.
A section of a function is simply a right inverse, by definition:
has-section : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ has-section r = Σ s ꞉ (codomain r → domain r), r ∘ s ∼ id
Notice that has-section r is the type of all sections (s , η) of
r, which may well be empty. So a point of this type is a designated
section s of r, together with the datum η. Unless the domain of
r is a set, this datum is not property, and we may well have an
element (s , η') of the type has-section r with η' distinct from
η for the same s.
We say that X is a retract of Y, written X ◁ Y, if we
have a function Y → X which has a section:
_◁_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ X ◁ Y = Σ r ꞉ (Y → X), has-section r
This type actually collects all the ways in which the type X can
be a retract of the type Y, and so is data or structure on X and
Y, rather than a property of them.
A function that has a section is called a retraction. We use this terminology, ambiguously, also for the function that projects out the retraction:
retraction : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ◁ Y → Y → X retraction (r , s , η) = r section : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ◁ Y → X → Y section (r , s , η) = s retract-equation : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (ρ : X ◁ Y) → retraction ρ ∘ section ρ ∼ 𝑖𝑑 X retract-equation (r , s , η) = η retraction-has-section : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (ρ : X ◁ Y) → has-section (retraction ρ) retraction-has-section (r , h) = h
We have an identity retraction:
id-◁ : (X : 𝓤 ̇ ) → X ◁ X id-◁ X = 𝑖𝑑 X , 𝑖𝑑 X , refl
Exercise. The identity retraction is by no means the only retraction
of a type onto itself in general, of course. Prove that we have (that
is, produce an element of the type) ℕ ◁ ℕ with the function
pred : ℕ → ℕ defined above as the retraction.
Try to produce more inhabitants of this type.
We can define the composition of two retractions as follows:
_◁∘_ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } → X ◁ Y → Y ◁ Z → X ◁ Z (r , s , η) ◁∘ (r' , s' , η') = (r ∘ r' , s' ∘ s , η'') where η'' = λ x → r (r' (s' (s x))) ＝⟨ ap r (η' (s x)) ⟩ r (s x) ＝⟨ η x ⟩ x ∎
We also define composition with an implicit argument made explicit:
_◁⟨_⟩_ : (X : 𝓤 ̇ ) {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } → X ◁ Y → Y ◁ Z → X ◁ Z X ◁⟨ ρ ⟩ σ = ρ ◁∘ σ
And we introduce the following postfix notation for the identity retraction:
_◀ : (X : 𝓤 ̇ ) → X ◁ X X ◀ = id-◁ X
These last two definitions are for notational convenience. See below for examples of their use.
We conclude this section with some facts about retracts of Σ types,
which are of general use, in particular for dealing with equivalences
in the sense of Voevosky in comparison with invertible
maps.
A pointwise retraction gives a retraction of the total spaces:
Σ-retract : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → ((x : X) → A x ◁ B x) → Σ A ◁ Σ B Σ-retract {𝓤} {𝓥} {𝓦} {X} {A} {B} ρ = NatΣ r , NatΣ s , η' where r : (x : X) → B x → A x r x = retraction (ρ x) s : (x : X) → A x → B x s x = section (ρ x) η : (x : X) (a : A x) → r x (s x a) ＝ a η x = retract-equation (ρ x) η' : (σ : Σ A) → NatΣ r (NatΣ s σ) ＝ σ η' (x , a) = x , r x (s x a) ＝⟨ to-Σ-＝' (η x a) ⟩ x , a ∎
We have that transport A (p ⁻¹) is a two-sided inverse of transport
A p using the functoriality of transport A, or directly by
induction on p:
transport-is-retraction : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} (p : x ＝ y) → transport A p ∘ transport A (p ⁻¹) ∼ 𝑖𝑑 (A y) transport-is-retraction A (refl x) = refl transport-is-section : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} (p : x ＝ y) → transport A (p ⁻¹) ∘ transport A p ∼ 𝑖𝑑 (A x) transport-is-section A (refl x) = refl
Using this, we have the following reindexing retraction of Σ types:
Σ-reindexing-retract : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : X → 𝓦 ̇ } (r : Y → X) → has-section r → (Σ x ꞉ X , A x) ◁ (Σ y ꞉ Y , A (r y)) Σ-reindexing-retract {𝓤} {𝓥} {𝓦} {X} {Y} {A} r (s , η) = γ , φ , γφ where γ : Σ (A ∘ r) → Σ A γ (y , a) = (r y , a) φ : Σ A → Σ (A ∘ r) φ (x , a) = (s x , transport A ((η x)⁻¹) a) γφ : (σ : Σ A) → γ (φ σ) ＝ σ γφ (x , a) = p where p : (r (s x) , transport A ((η x)⁻¹) a) ＝ (x , a) p = to-Σ-＝ (η x , transport-is-retraction A (η x) a)
We have defined the property of a type being a
singleton. The singleton type Σ y ꞉ X ,
x ＝ y induced by a point x : X of a type X is denoted by
singleton-type x. The terminology is justified by the fact that it
is indeed a singleton in the sense defined above.
singleton-type : {X : 𝓤 ̇ } → X → 𝓤 ̇ singleton-type {𝓤} {X} x = Σ y ꞉ X , y ＝ x singleton-type-center : {X : 𝓤 ̇ } (x : X) → singleton-type x singleton-type-center x = (x , refl x) singleton-type-centered : {X : 𝓤 ̇ } (x : X) (σ : singleton-type x) → singleton-type-center x ＝ σ singleton-type-centered x (x , refl x) = refl (x , refl x) singleton-types-are-singletons : (X : 𝓤 ̇ ) (x : X) → is-singleton (singleton-type x) singleton-types-are-singletons X x = singleton-type-center x , singleton-type-centered x
The following gives a technique for showing that some types are singletons:
retract-of-singleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → Y ◁ X → is-singleton X → is-singleton Y retract-of-singleton (r , s , η) (c , φ) = r c , γ where γ = λ y → r c ＝⟨ ap r (φ (s y)) ⟩ r (s y) ＝⟨ η y ⟩ y ∎
Sometimes we need the following symmetric versions of the above:
singleton-type' : {X : 𝓤 ̇ } → X → 𝓤 ̇ singleton-type' {𝓤} {X} x = Σ y ꞉ X , x ＝ y singleton-type'-center : {X : 𝓤 ̇ } (x : X) → singleton-type' x singleton-type'-center x = (x , refl x) singleton-type'-centered : {X : 𝓤 ̇ } (x : X) (σ : singleton-type' x) → singleton-type'-center x ＝ σ singleton-type'-centered x (x , refl x) = refl (x , refl x) singleton-types'-are-singletons : (X : 𝓤 ̇ ) (x : X) → is-singleton (singleton-type' x) singleton-types'-are-singletons X x = singleton-type'-center x , singleton-type'-centered x
The main notions of univalent mathematics conceived by Voevodsky, with formulations in MLTT, are those of singleton type (or contractible type), hlevel (including the notions of subsingleton and set), and of type equivalence, which we define now.
We begin with a discussion of the notion of invertible function, whose only difference with the notion of equivalence is that it is data rather than property:
invertible : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ invertible f = Σ g ꞉ (codomain f → domain f) , (g ∘ f ∼ id) × (f ∘ g ∼ id)
The situation is that we will have a logical equivalence between
data establishing invertibility of a given function, and
the property of the function being an equivalence.
Mathematically, what happens is that the type
f is an equivalenceis a retract of the type
f is invertible.This retraction property is not easy to show, and there are many approaches. We discuss an approach we came up with while developing these lecture notes, which is intended to be relatively simple and direct, but the reader should consult other approaches, such as that of the HoTT book, which has a well-established categorical pedigree.
The problem with the notion of invertibility of f is that, while we
have that the inverse g is unique when it exists, we cannot in
general prove that the identification data g ∘ f ∼ id and f ∘ g ∼
id are also unique, and, indeed, they are not in
general.
The following is Voevodsky’s proposed formulation of the notion of
equivalence in MLTT, which relies on the concept of fiber:
fiber : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → Y → 𝓤 ⊔ 𝓥 ̇ fiber f y = Σ x ꞉ domain f , f x ＝ y fiber-point : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {f : X → Y} {y : Y} → fiber f y → X fiber-point (x , p) = x fiber-identification : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {f : X → Y} {y : Y} → (w : fiber f y) → f (fiber-point w) ＝ y fiber-identification (x , p) = p
So the type fiber f y collects the points x : X which are mapped
to a point identified with y, including the identification
datum. Voevodsky’s insight is that a general notion of equivalence can
be formulated in MLTT by requiring the fibers to be singletons. It is
important here that not only the x : X with f x ＝ y is unique, but
also that the identification datum p : f x ＝ y is unique. This is
similar to, or even a generalization, of the categorical
notion of “uniqueness up to a unique isomorphism”.
is-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ is-equiv f = (y : codomain f) → is-singleton (fiber f y)
We can read this as saying that for every y : Y there is a unique x : X with f x ＝ y, where the uniqueness refers not only to x : X
but also to the identification datum p : f x ＝ y. More precisely,
the pair (x , p) is required to be unique. It is easy to see that
equivalences are invertible:
inverse : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → (Y → X) inverse f e y = fiber-point (center (fiber f y) (e y)) inverses-are-sections : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (e : is-equiv f) → f ∘ inverse f e ∼ id inverses-are-sections f e y = fiber-identification (center (fiber f y) (e y)) inverse-centrality : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (e : is-equiv f) (y : Y) (t : fiber f y) → (inverse f e y , inverses-are-sections f e y) ＝ t inverse-centrality f e y = centrality (fiber f y) (e y) inverses-are-retractions : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (e : is-equiv f) → inverse f e ∘ f ∼ id inverses-are-retractions f e x = ap fiber-point p where p : inverse f e (f x) , inverses-are-sections f e (f x) ＝ x , refl (f x) p = inverse-centrality f e (f x) (x , (refl (f x))) equivs-are-invertible : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → invertible f equivs-are-invertible f e = inverse f e , inverses-are-retractions f e , inverses-are-sections f e
The non-trivial direction derives the equivalence property from invertibility data, for which we use the retraction techniques explained above.
Suppose that invertibility data
g : Y → X,
η : (x : X) → g (f x) ＝ x
ε : (y : Y) → f (g y) ＝ y
for a map f : X → Y is given, and that a point y₀ in the codomain
of f is given.
We need to show that the fiber Σ x ꞉ X , f x ＝ y₀ of y₀ is a
singleton.
We first use the assumption ε to show that the type f (g y) ＝
y₀ is a retract of the type y ＝ y₀ for any given y : Y.
To get the section s : f (g y) ＝ y₀ → y ＝ y₀, we transport along
the identification ε y : f (g y) ＝ y over the family A - = (-
＝ y₀), which can be abbreviated as _＝ y₀.
To get the retraction r in the opposite direction, we transport
along the inverse of the identification ε y over the same
family.
We already know that this gives a section-retraction pair by
transport-is-section.
Next we have that the type Σ x ꞉ X , f x ＝ y₀ is a retract
of the type Σ y ꞉ Y , f (g y) ＝ y₀ by Σ-reindexing-retract
using the assumption that η exibits g as a section of f,
which in turn is a retract of the type Σ y ꞉ Y , y ＝ y₀ by
applying Σ to both sides of the retraction (f (g y) ＝ y₀) ◁ (y
＝ y₀) of the previous step.
This amounts to saying that the type fiber f y₀ is a retract of
singleton-type y₀.
But then we are done, because singleton types are singletons and retracts of singletons are singletons.
invertibles-are-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → invertible f → is-equiv f invertibles-are-equivs {𝓤} {𝓥} {X} {Y} f (g , η , ε) y₀ = iii where i : (y : Y) → (f (g y) ＝ y₀) ◁ (y ＝ y₀) i y = r , s , transport-is-section (_＝ y₀) (ε y) where s : f (g y) ＝ y₀ → y ＝ y₀ s = transport (_＝ y₀) (ε y) r : y ＝ y₀ → f (g y) ＝ y₀ r = transport (_＝ y₀) ((ε y)⁻¹) ii : fiber f y₀ ◁ singleton-type y₀ ii = (Σ x ꞉ X , f x ＝ y₀) ◁⟨ Σ-reindexing-retract g (f , η) ⟩ (Σ y ꞉ Y , f (g y) ＝ y₀) ◁⟨ Σ-retract i ⟩ (Σ y ꞉ Y , y ＝ y₀) ◀ iii : is-singleton (fiber f y₀) iii = retract-of-singleton ii (singleton-types-are-singletons Y y₀)
An immediate consequence is that inverses of equivalences are themselves equivalences:
inverses-are-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (e : is-equiv f) → is-equiv (inverse f e) inverses-are-equivs f e = invertibles-are-equivs (inverse f e) (f , inverses-are-sections f e , inverses-are-retractions f e)
Notice that inversion is involutive on the nose:
inversion-involutive : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (e : is-equiv f) → inverse (inverse f e) (inverses-are-equivs f e) ＝ f inversion-involutive f e = refl f
To see that the above procedures do exhibit the type “f is an
equivalence” as a retract of the type “f is invertible”, it suffices
to show that it is a
subsingleton.
The identity function is invertible:
id-invertible : (X : 𝓤 ̇ ) → invertible (𝑖𝑑 X) id-invertible X = 𝑖𝑑 X , refl , refl
We can compose invertible maps:
∘-invertible : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } {f : X → Y} {f' : Y → Z} → invertible f' → invertible f → invertible (f' ∘ f) ∘-invertible {𝓤} {𝓥} {𝓦} {X} {Y} {Z} {f} {f'} (g' , gf' , fg') (g , gf , fg) = g ∘ g' , η , ε where η = λ x → g (g' (f' (f x))) ＝⟨ ap g (gf' (f x)) ⟩ g (f x) ＝⟨ gf x ⟩ x ∎ ε = λ z → f' (f (g (g' z))) ＝⟨ ap f' (fg (g' z)) ⟩ f' (g' z) ＝⟨ fg' z ⟩ z ∎
There is an identity equivalence, and we get composition of equivalences by reduction to invertible maps:
id-is-equiv : (X : 𝓤 ̇ ) → is-equiv (𝑖𝑑 X) id-is-equiv = singleton-types-are-singletons
An abstract definition is not expanded during type checking. One
possible use of this is efficiency. In our case, it saves about half a
minute in the checking of this file for correctness in the uses of
∘-is-equiv:
∘-is-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } {f : X → Y} {g : Y → Z} → is-equiv g → is-equiv f → is-equiv (g ∘ f) ∘-is-equiv {𝓤} {𝓥} {𝓦} {X} {Y} {Z} {f} {g} i j = γ where abstract γ : is-equiv (g ∘ f) γ = invertibles-are-equivs (g ∘ f) (∘-invertible (equivs-are-invertible g i) (equivs-are-invertible f j))
Because we have made the above definition abstract, we don’t have
access to the given construction when proving things involving
∘-is-equiv, such as the contravariance of inversion:
inverse-of-∘ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) (g : Y → Z) (i : is-equiv f) (j : is-equiv g) → inverse f i ∘ inverse g j ∼ inverse (g ∘ f) (∘-is-equiv j i) inverse-of-∘ f g i j z = f' (g' z) ＝⟨ (ap (f' ∘ g') (s z))⁻¹ ⟩ f' (g' (g (f (h z)))) ＝⟨ ap f' (inverses-are-retractions g j (f (h z))) ⟩ f' (f (h z)) ＝⟨ inverses-are-retractions f i (h z) ⟩ h z ∎ where f' = inverse f i g' = inverse g j h = inverse (g ∘ f) (∘-is-equiv j i) s : g ∘ f ∘ h ∼ id s = inverses-are-sections (g ∘ f) (∘-is-equiv j i)
The type of equivalences is defined as follows:
_≃_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ X ≃ Y = Σ f ꞉ (X → Y), is-equiv f
Notice that this doesn’t just say that X and Y are equivalent: the
type X ≃ Y collects all the ways in which the types X and Y are
equivalent. For example, the two-point type 𝟚 is equivalent to
itself in two ways, by the identity map, and by the map that
interchanges its two points, and hence the type 𝟚 ≃ 𝟚 has two
elements.
Again it is convenient to have special names for its first and second projections:
Eq→fun ⌜_⌝ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → X → Y Eq→fun (f , i) = f ⌜_⌝ = Eq→fun Eq→fun-is-equiv ⌜⌝-is-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (e : X ≃ Y) → is-equiv (⌜ e ⌝) Eq→fun-is-equiv (f , i) = i ⌜⌝-is-equiv = Eq→fun-is-equiv invertibility-gives-≃ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → invertible f → X ≃ Y invertibility-gives-≃ f i = f , invertibles-are-equivs f i
Examples:
Σ-induction-≃ : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {A : Σ Y → 𝓦 ̇ } → ((x : X) (y : Y x) → A (x , y)) ≃ ((z : Σ Y) → A z) Σ-induction-≃ = invertibility-gives-≃ Σ-induction (curry , refl , refl) Σ-flip : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : X → Y → 𝓦 ̇ } → (Σ x ꞉ X , Σ y ꞉ Y , A x y) ≃ (Σ y ꞉ Y , Σ x ꞉ X , A x y) Σ-flip = invertibility-gives-≃ (λ (x , y , p) → (y , x , p)) ((λ (y , x , p) → (x , y , p)) , refl , refl) ×-comm : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X × Y) ≃ (Y × X) ×-comm = invertibility-gives-≃ (λ (x , y) → (y , x)) ((λ (y , x) → (x , y)) , refl , refl)
The identity equivalence and the composition of two equivalences:
id-≃ : (X : 𝓤 ̇ ) → X ≃ X id-≃ X = 𝑖𝑑 X , id-is-equiv X _●_ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } → X ≃ Y → Y ≃ Z → X ≃ Z (f , d) ● (f' , e) = f' ∘ f , ∘-is-equiv e d ≃-sym : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → Y ≃ X ≃-sym (f , e) = inverse f e , inverses-are-equivs f e
We can use the following notation for equational reasoning with equivalences:
_≃⟨_⟩_ : (X : 𝓤 ̇ ) {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } → X ≃ Y → Y ≃ Z → X ≃ Z _ ≃⟨ d ⟩ e = d ● e _■ : (X : 𝓤 ̇ ) → X ≃ X _■ = id-≃
We conclude this section with some important examples.
The function transport A p is an equivalence.
transport-is-equiv : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) {x y : X} (p : x ＝ y) → is-equiv (transport A p) transport-is-equiv A (refl x) = id-is-equiv (A x)
Alternatively, we could have used the fact that transport A (p ⁻¹)
is an inverse of transport A p.
Here is the promised characterization of equality in Σ types:
Σ-＝-≃ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (σ τ : Σ A) → (σ ＝ τ) ≃ (Σ p ꞉ pr₁ σ ＝ pr₁ τ , transport A p (pr₂ σ) ＝ pr₂ τ) Σ-＝-≃ {𝓤} {𝓥} {X} {A} σ τ = invertibility-gives-≃ from-Σ-＝ (to-Σ-＝ , η , ε) where η : (q : σ ＝ τ) → to-Σ-＝ (from-Σ-＝ q) ＝ q η (refl σ) = refl (refl σ) ε : (w : Σ p ꞉ pr₁ σ ＝ pr₁ τ , transport A p (pr₂ σ) ＝ pr₂ τ) → from-Σ-＝ (to-Σ-＝ w) ＝ w ε (refl p , refl q) = refl (refl p , refl q)
Similarly we have:
to-×-＝ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {z t : X × Y} → (pr₁ z ＝ pr₁ t) × (pr₂ z ＝ pr₂ t) → z ＝ t to-×-＝ (refl x , refl y) = refl (x , y) from-×-＝ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {z t : X × Y} → z ＝ t → (pr₁ z ＝ pr₁ t) × (pr₂ z ＝ pr₂ t) from-×-＝ {𝓤} {𝓥} {X} {Y} (refl (x , y)) = (refl x , refl y) ×-＝-≃ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (z t : X × Y) → (z ＝ t) ≃ (pr₁ z ＝ pr₁ t) × (pr₂ z ＝ pr₂ t) ×-＝-≃ {𝓤} {𝓥} {X} {Y} z t = invertibility-gives-≃ from-×-＝ (to-×-＝ , η , ε) where η : (p : z ＝ t) → to-×-＝ (from-×-＝ p) ＝ p η (refl z) = refl (refl z) ε : (q : (pr₁ z ＝ pr₁ t) × (pr₂ z ＝ pr₂ t)) → from-×-＝ (to-×-＝ q) ＝ q ε (refl x , refl y) = refl (refl x , refl y)
The following are often useful:
ap-pr₁-to-×-＝ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {z t : X × Y} → (p₁ : pr₁ z ＝ pr₁ t) → (p₂ : pr₂ z ＝ pr₂ t) → ap pr₁ (to-×-＝ (p₁ , p₂)) ＝ p₁ ap-pr₁-to-×-＝ (refl x) (refl y) = refl (refl x) ap-pr₂-to-×-＝ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {z t : X × Y} → (p₁ : pr₁ z ＝ pr₁ t) → (p₂ : pr₂ z ＝ pr₂ t) → ap pr₂ (to-×-＝ (p₁ , p₂)) ＝ p₂ ap-pr₂-to-×-＝ (refl x) (refl y) = refl (refl y) Σ-cong : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → ((x : X) → A x ≃ B x) → Σ A ≃ Σ B Σ-cong {𝓤} {𝓥} {𝓦} {X} {A} {B} φ = invertibility-gives-≃ (NatΣ f) (NatΣ g , NatΣ-η , NatΣ-ε) where f : (x : X) → A x → B x f x = ⌜ φ x ⌝ g : (x : X) → B x → A x g x = inverse (f x) (⌜⌝-is-equiv (φ x)) η : (x : X) (a : A x) → g x (f x a) ＝ a η x = inverses-are-retractions (f x) (⌜⌝-is-equiv (φ x)) ε : (x : X) (b : B x) → f x (g x b) ＝ b ε x = inverses-are-sections (f x) (⌜⌝-is-equiv (φ x)) NatΣ-η : (w : Σ A) → NatΣ g (NatΣ f w) ＝ w NatΣ-η (x , a) = x , g x (f x a) ＝⟨ to-Σ-＝' (η x a) ⟩ x , a ∎ NatΣ-ε : (t : Σ B) → NatΣ f (NatΣ g t) ＝ t NatΣ-ε (x , b) = x , f x (g x b) ＝⟨ to-Σ-＝' (ε x b) ⟩ x , b ∎ ≃-gives-◁ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → X ◁ Y ≃-gives-◁ (f , e) = (inverse f e , f , inverses-are-retractions f e) ≃-gives-▷ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → Y ◁ X ≃-gives-▷ (f , e) = (f , inverse f e , inverses-are-sections f e) equiv-to-singleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-singleton Y → is-singleton X equiv-to-singleton e = retract-of-singleton (≃-gives-◁ e)
There is a canonical transformation (X Y : 𝓤 ̇ ) → X ＝ Y → X ≃ Y that
sends the identity identification refl X : X ＝ X to the identity
equivalence id-≃ X : X ≃ X. The univalence axiom, for the universe
𝓤, says that this canonical map is itself an equivalence.
Id→Eq : (X Y : 𝓤 ̇ ) → X ＝ Y → X ≃ Y Id→Eq X X (refl X) = id-≃ X is-univalent : (𝓤 : Universe) → 𝓤 ⁺ ̇ is-univalent 𝓤 = (X Y : 𝓤 ̇ ) → is-equiv (Id→Eq X Y)
Thus, the univalence of the universe 𝓤 says that identifications X
＝ Y of types in 𝓤 are in canonical bijection with equivalences X ≃ Y, if by
bijection we mean equivalence,
where the canonical bijection is
Id→Eq.
We emphasize that this doesn’t posit that univalence holds. It says what univalence is (like the type that says what the twin-prime conjecture is).
univalence-≃ : is-univalent 𝓤 → (X Y : 𝓤 ̇ ) → (X ＝ Y) ≃ (X ≃ Y) univalence-≃ ua X Y = Id→Eq X Y , ua X Y Eq→Id : is-univalent 𝓤 → (X Y : 𝓤 ̇ ) → X ≃ Y → X ＝ Y Eq→Id ua X Y = inverse (Id→Eq X Y) (ua X Y)
Here is a third way to convert a type identification into a function:
Id→fun : {X Y : 𝓤 ̇ } → X ＝ Y → X → Y Id→fun {𝓤} {X} {Y} p = ⌜ Id→Eq X Y p ⌝ Id→funs-agree : {X Y : 𝓤 ̇ } (p : X ＝ Y) → Id→fun p ＝ Id→Fun p Id→funs-agree (refl X) = refl (𝑖𝑑 X)
What characterizes univalent mathematics is not the univalence axiom. We have defined and studied the main concepts of univalent mathematics in a pure, spartan MLTT. It is the concepts of hlevel, including singleton, subsingleton and set, and the notion of equivalence that are at the heart of univalent mathematics. Univalence is a fundamental ingredient, but first we need the correct notion of equivalence to be able to formulate it.
Remark. If we formulate univalence with invertible maps instead of equivalences, we get a statement that is provably false in MLTT, and this is one of the reasons why Voevodsky’s notion of equivalence is important. This is Exercise 4.6 of the HoTT book. There is a solution in Coq by Mike Shulman.
We have two automorphisms of 𝟚, namely the identity function and the
map that swaps ₀ and ₁:
swap₂ : 𝟚 → 𝟚 swap₂ ₀ = ₁ swap₂ ₁ = ₀ swap₂-involutive : (n : 𝟚) → swap₂ (swap₂ n) ＝ n swap₂-involutive ₀ = refl ₀ swap₂-involutive ₁ = refl ₁
That is, swap₂ is its own inverse and hence is an equivalence:
swap₂-is-equiv : is-equiv swap₂ swap₂-is-equiv = invertibles-are-equivs swap₂ (swap₂ , swap₂-involutive , swap₂-involutive)
We now use a local module to assume univalence of the first universe in the construction of our example:
module example-of-a-nonset (ua : is-univalent 𝓤₀) where
The above gives two distinct equivalences:
e₀ e₁ : 𝟚 ≃ 𝟚 e₀ = id-≃ 𝟚 e₁ = swap₂ , swap₂-is-equiv e₀-is-not-e₁ : e₀ ≠ e₁ e₀-is-not-e₁ p = ₁-is-not-₀ r where q : id ＝ swap₂ q = ap ⌜_⌝ p r : ₁ ＝ ₀ r = ap (λ - → - ₁) q
Using univalence, we get two different identifications of the type
𝟚 with itself:
p₀ p₁ : 𝟚 ＝ 𝟚 p₀ = Eq→Id ua 𝟚 𝟚 e₀ p₁ = Eq→Id ua 𝟚 𝟚 e₁ p₀-is-not-p₁ : p₀ ≠ p₁ p₀-is-not-p₁ q = e₀-is-not-e₁ r where r = e₀ ＝⟨ (inverses-are-sections (Id→Eq 𝟚 𝟚) (ua 𝟚 𝟚) e₀)⁻¹ ⟩ Id→Eq 𝟚 𝟚 p₀ ＝⟨ ap (Id→Eq 𝟚 𝟚) q ⟩ Id→Eq 𝟚 𝟚 p₁ ＝⟨ inverses-are-sections (Id→Eq 𝟚 𝟚) (ua 𝟚 𝟚) e₁ ⟩ e₁ ∎
If the universe 𝓤₀ were a set, then the identifications p₀ and
p₁ defined above would be equal, and therefore it is not a set.
𝓤₀-is-not-a-set : ¬ (is-set (𝓤₀ ̇ )) 𝓤₀-is-not-a-set s = p₀-is-not-p₁ q where q : p₀ ＝ p₁ q = s 𝟚 𝟚 p₀ p₁
For more examples, see [Kraus and Sattler].
Here are some facts whose proofs are left to the reader but that we will need from the next section onwards. Sample solutions are given below.
Define functions for the following type declarations. As a matter of
procedure, we suggest to import this file in a solutions file and add
another declaration with the same type and new name
e.g. sections-are-lc-solution, because we already have solutions in
this file. It is important not to forget to include the option
--without-K in the solutions file that imports (the Agda version of)
this file.
subsingleton-criterion : {X : 𝓤 ̇ } → (X → is-singleton X) → is-subsingleton X subsingleton-criterion' : {X : 𝓤 ̇ } → (X → is-subsingleton X) → is-subsingleton X retract-of-subsingleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → Y ◁ X → is-subsingleton X → is-subsingleton Y left-cancellable : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ left-cancellable f = {x x' : domain f} → f x ＝ f x' → x ＝ x' lc-maps-reflect-subsingletons : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → left-cancellable f → is-subsingleton Y → is-subsingleton X has-retraction : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ has-retraction s = Σ r ꞉ (codomain s → domain s), r ∘ s ∼ id sections-are-lc : {X : 𝓤 ̇ } {A : 𝓥 ̇ } (s : X → A) → has-retraction s → left-cancellable s equivs-have-retractions : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → has-retraction f equivs-have-sections : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → has-section f equivs-are-lc : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → left-cancellable f equiv-to-subsingleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-subsingleton Y → is-subsingleton X comp-inverses : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) (g : Y → Z) (i : is-equiv f) (j : is-equiv g) (f' : Y → X) (g' : Z → Y) → f' ∼ inverse f i → g' ∼ inverse g j → f' ∘ g' ∼ inverse (g ∘ f) (∘-is-equiv j i) equiv-to-set : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-set Y → is-set X sections-closed-under-∼ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f g : X → Y) → has-retraction f → g ∼ f → has-retraction g retractions-closed-under-∼ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f g : X → Y) → has-section f → g ∼ f → has-section g
An alternative notion of equivalence, equivalent to Voevodsky’s, has been given by André Joyal:
is-joyal-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ is-joyal-equiv f = has-section f × has-retraction f
Provide definitions for the following type declarations:
one-inverse : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) (f : X → Y) (r s : Y → X) → (r ∘ f ∼ id) → (f ∘ s ∼ id) → r ∼ s joyal-equivs-are-invertible : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-joyal-equiv f → invertible f joyal-equivs-are-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-joyal-equiv f → is-equiv f invertibles-are-joyal-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → invertible f → is-joyal-equiv f equivs-are-joyal-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → is-joyal-equiv f equivs-closed-under-∼ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {f g : X → Y} → is-equiv f → g ∼ f → is-equiv g equiv-to-singleton' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-singleton X → is-singleton Y subtypes-of-sets-are-sets : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (m : X → Y) → left-cancellable m → is-set Y → is-set X pr₁-lc : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-subsingleton (A x)) → left-cancellable (λ (t : Σ A) → pr₁ t) subsets-of-sets-are-sets : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) → is-set X → ((x : X) → is-subsingleton (A x)) → is-set (Σ x ꞉ X , A x) to-subtype-＝ : {X : 𝓦 ̇ } {A : X → 𝓥 ̇ } {x y : X} {a : A x} {b : A y} → ((x : X) → is-subsingleton (A x)) → x ＝ y → (x , a) ＝ (y , b) pr₁-is-equiv : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-singleton (A x)) → is-equiv (λ (t : Σ A) → pr₁ t) pr₁-≃ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-singleton (A x)) → Σ A ≃ X pr₁-≃ i = pr₁ , pr₁-is-equiv i ΠΣ-distr-≃ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {P : (x : X) → A x → 𝓦 ̇ } → (Π x ꞉ X , Σ a ꞉ A x , P x a) ≃ (Σ f ꞉ Π A , Π x ꞉ X , P x (f x)) Σ-assoc : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {Z : Σ Y → 𝓦 ̇ } → Σ Z ≃ (Σ x ꞉ X , Σ y ꞉ Y x , Z (x , y)) ⁻¹-≃ : {X : 𝓤 ̇ } (x y : X) → (x ＝ y) ≃ (y ＝ x) singleton-types-≃ : {X : 𝓤 ̇ } (x : X) → singleton-type' x ≃ singleton-type x singletons-≃ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-singleton X → is-singleton Y → X ≃ Y maps-of-singletons-are-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-singleton X → is-singleton Y → is-equiv f logically-equivalent-subsingletons-are-equivalent : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → is-subsingleton X → is-subsingleton Y → X ⇔ Y → X ≃ Y singletons-are-equivalent : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → is-singleton X → is-singleton Y → X ≃ Y NatΣ-fiber-equiv : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (B : X → 𝓦 ̇ ) (φ : Nat A B) (x : X) (b : B x) → fiber (φ x) b ≃ fiber (NatΣ φ) (x , b) NatΣ-equiv-gives-fiberwise-equiv : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } (φ : Nat A B) → is-equiv (NatΣ φ) → ((x : X) → is-equiv (φ x)) Σ-is-subsingleton : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → is-subsingleton X → ((x : X) → is-subsingleton (A x)) → is-subsingleton (Σ A) ×-is-singleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-singleton X → is-singleton Y → is-singleton (X × Y) ×-is-subsingleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-subsingleton X → is-subsingleton Y → is-subsingleton (X × Y) ×-is-subsingleton' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → ((Y → is-subsingleton X) × (X → is-subsingleton Y)) → is-subsingleton (X × Y) ×-is-subsingleton'-back : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-subsingleton (X × Y) → (Y → is-subsingleton X) × (X → is-subsingleton Y) ap₂ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y → Z) {x x' : X} {y y' : Y} → x ＝ x' → y ＝ y' → f x y ＝ f x' y'
For the sake of readability, we re-state the formulations of the
exercises in the type of sol in a where clause for each exercise.
subsingleton-criterion = sol where sol : {X : 𝓤 ̇ } → (X → is-singleton X) → is-subsingleton X sol f x = singletons-are-subsingletons (domain f) (f x) x subsingleton-criterion' = sol where sol : {X : 𝓤 ̇ } → (X → is-subsingleton X) → is-subsingleton X sol f x y = f x x y retract-of-subsingleton = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → Y ◁ X → is-subsingleton X → is-subsingleton Y sol (r , s , η) i = subsingleton-criterion (λ x → retract-of-singleton (r , s , η) (pointed-subsingletons-are-singletons (codomain s) (s x) i)) lc-maps-reflect-subsingletons = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → left-cancellable f → is-subsingleton Y → is-subsingleton X sol f l s x x' = l (s (f x) (f x')) sections-are-lc = sol where sol : {X : 𝓤 ̇ } {A : 𝓥 ̇ } (s : X → A) → has-retraction s → left-cancellable s sol s (r , ε) {x} {y} p = x ＝⟨ (ε x)⁻¹ ⟩ r (s x) ＝⟨ ap r p ⟩ r (s y) ＝⟨ ε y ⟩ y ∎ equivs-have-retractions = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → has-retraction f sol f e = (inverse f e , inverses-are-retractions f e) equivs-have-sections = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → has-section f sol f e = (inverse f e , inverses-are-sections f e) equivs-are-lc = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → left-cancellable f sol f e = sections-are-lc f (equivs-have-retractions f e) equiv-to-subsingleton = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-subsingleton Y → is-subsingleton X sol (f , i) = lc-maps-reflect-subsingletons f (equivs-are-lc f i) comp-inverses = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) (g : Y → Z) (i : is-equiv f) (j : is-equiv g) (f' : Y → X) (g' : Z → Y) → f' ∼ inverse f i → g' ∼ inverse g j → f' ∘ g' ∼ inverse (g ∘ f) (∘-is-equiv j i) sol f g i j f' g' h k z = f' (g' z) ＝⟨ h (g' z) ⟩ inverse f i (g' z) ＝⟨ ap (inverse f i) (k z) ⟩ inverse f i (inverse g j z) ＝⟨ inverse-of-∘ f g i j z ⟩ inverse (g ∘ f) (∘-is-equiv j i) z ∎ equiv-to-set = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-set Y → is-set X sol e = subtypes-of-sets-are-sets ⌜ e ⌝ (equivs-are-lc ⌜ e ⌝ (⌜⌝-is-equiv e)) sections-closed-under-∼ = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f g : X → Y) → has-retraction f → g ∼ f → has-retraction g sol f g (r , rf) h = (r , λ x → r (g x) ＝⟨ ap r (h x) ⟩ r (f x) ＝⟨ rf x ⟩ x ∎) retractions-closed-under-∼ = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f g : X → Y) → has-section f → g ∼ f → has-section g sol f g (s , fs) h = (s , λ y → g (s y) ＝⟨ h (s y) ⟩ f (s y) ＝⟨ fs y ⟩ y ∎) one-inverse = sol where sol : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) (f : X → Y) (r s : Y → X) → (r ∘ f ∼ id) → (f ∘ s ∼ id) → r ∼ s sol X Y f r s h k y = r y ＝⟨ ap r ((k y)⁻¹) ⟩ r (f (s y)) ＝⟨ h (s y) ⟩ s y ∎ joyal-equivs-are-invertible = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-joyal-equiv f → invertible f sol f ((s , ε) , (r , η)) = (s , sf , ε) where sf = λ (x : domain f) → s(f x) ＝⟨ (η (s (f x)))⁻¹ ⟩ r(f(s(f x))) ＝⟨ ap r (ε (f x)) ⟩ r(f x) ＝⟨ η x ⟩ x ∎ joyal-equivs-are-equivs = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-joyal-equiv f → is-equiv f sol f j = invertibles-are-equivs f (joyal-equivs-are-invertible f j) invertibles-are-joyal-equivs = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → invertible f → is-joyal-equiv f sol f (g , η , ε) = ((g , ε) , (g , η)) equivs-are-joyal-equivs = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → is-joyal-equiv f sol f e = invertibles-are-joyal-equivs f (equivs-are-invertible f e) equivs-closed-under-∼ = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {f g : X → Y} → is-equiv f → g ∼ f → is-equiv g sol {𝓤} {𝓥} {X} {Y} {f} {g} e h = joyal-equivs-are-equivs g (retractions-closed-under-∼ f g (equivs-have-sections f e) h , sections-closed-under-∼ f g (equivs-have-retractions f e) h) equiv-to-singleton' = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → X ≃ Y → is-singleton X → is-singleton Y sol e = retract-of-singleton (≃-gives-▷ e) subtypes-of-sets-are-sets = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (m : X → Y) → left-cancellable m → is-set Y → is-set X sol {𝓤} {𝓥} {X} m i h = types-with-wconstant-＝-endomaps-are-sets X c where f : (x x' : X) → x ＝ x' → x ＝ x' f x x' r = i (ap m r) κ : (x x' : X) (r s : x ＝ x') → f x x' r ＝ f x x' s κ x x' r s = ap i (h (m x) (m x') (ap m r) (ap m s)) c : wconstant-＝-endomaps X c x x' = f x x' , κ x x' pr₁-lc = sol where sol : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-subsingleton (A x)) → left-cancellable (λ (t : Σ A) → pr₁ t) sol i p = to-Σ-＝ (p , i _ _ _) subsets-of-sets-are-sets = sol where sol : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) → is-set X → ((x : X) → is-subsingleton (A x)) → is-set (Σ x ꞉ X , A x) sol X A h p = subtypes-of-sets-are-sets pr₁ (pr₁-lc p) h to-subtype-＝ = sol where sol : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {x y : X} {a : A x} {b : A y} → ((x : X) → is-subsingleton (A x)) → x ＝ y → (x , a) ＝ (y , b) sol {𝓤} {𝓥} {X} {A} {x} {y} {a} {b} s p = to-Σ-＝ (p , s y (transport A p a) b) pr₁-is-equiv = sol where sol : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-singleton (A x)) → is-equiv (λ (t : Σ A) → pr₁ t) sol {𝓤} {𝓥} {X} {A} s = invertibles-are-equivs pr₁ (g , η , ε) where g : X → Σ A g x = x , pr₁(s x) ε : (x : X) → pr₁ (g x) ＝ x ε x = refl (pr₁ (g x)) η : (σ : Σ A) → g (pr₁ σ) ＝ σ η (x , a) = to-subtype-＝ (λ x → singletons-are-subsingletons (A x) (s x)) (ε x) ΠΣ-distr-≃ = sol where sol : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {P : (x : X) → A x → 𝓦 ̇ } → (Π x ꞉ X , Σ a ꞉ A x , P x a) ≃ (Σ f ꞉ Π A , Π x ꞉ X , P x (f x)) sol {𝓤} {𝓥} {𝓦} {X} {A} {P} = invertibility-gives-≃ φ (γ , η , ε) where φ : (Π x ꞉ X , Σ a ꞉ A x , P x a) → Σ f ꞉ Π A , Π x ꞉ X , P x (f x) φ g = ((λ x → pr₁ (g x)) , λ x → pr₂ (g x)) γ : (Σ f ꞉ Π A , Π x ꞉ X , P x (f x)) → Π x ꞉ X , Σ a ꞉ A x , P x a γ (f , φ) x = f x , φ x η : γ ∘ φ ∼ id η = refl ε : φ ∘ γ ∼ id ε = refl Σ-assoc = sol where sol : {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {Z : Σ Y → 𝓦 ̇ } → Σ Z ≃ (Σ x ꞉ X , Σ y ꞉ Y x , Z (x , y)) sol {𝓤} {𝓥} {𝓦} {X} {Y} {Z} = invertibility-gives-≃ f (g , refl , refl) where f : Σ Z → Σ x ꞉ X , Σ y ꞉ Y x , Z (x , y) f ((x , y) , z) = (x , (y , z)) g : (Σ x ꞉ X , Σ y ꞉ Y x , Z (x , y)) → Σ Z g (x , (y , z)) = ((x , y) , z) ⁻¹-is-equiv : {X : 𝓤 ̇ } (x y : X) → is-equiv (λ (p : x ＝ y) → p ⁻¹) ⁻¹-is-equiv x y = invertibles-are-equivs _⁻¹ (_⁻¹ , ⁻¹-involutive , ⁻¹-involutive) ⁻¹-≃ = sol where sol : {X : 𝓤 ̇ } (x y : X) → (x ＝ y) ≃ (y ＝ x) sol x y = (_⁻¹ , ⁻¹-is-equiv x y) singleton-types-≃ = sol where sol : {X : 𝓤 ̇ } (x : X) → singleton-type' x ≃ singleton-type x sol x = Σ-cong (λ y → ⁻¹-≃ x y) singletons-≃ = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-singleton X → is-singleton Y → X ≃ Y sol {𝓤} {𝓥} {X} {Y} i j = invertibility-gives-≃ f (g , η , ε) where f : X → Y f x = center Y j g : Y → X g y = center X i η : (x : X) → g (f x) ＝ x η = centrality X i ε : (y : Y) → f (g y) ＝ y ε = centrality Y j maps-of-singletons-are-equivs = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-singleton X → is-singleton Y → is-equiv f sol {𝓤} {𝓥} {X} {Y} f i j = invertibles-are-equivs f (g , η , ε) where g : Y → X g y = center X i η : (x : X) → g (f x) ＝ x η = centrality X i ε : (y : Y) → f (g y) ＝ y ε y = singletons-are-subsingletons Y j (f (g y)) y logically-equivalent-subsingletons-are-equivalent = sol where sol : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → is-subsingleton X → is-subsingleton Y → X ⇔ Y → X ≃ Y sol X Y i j (f , g) = invertibility-gives-≃ f (g , (λ x → i (g (f x)) x) , (λ y → j (f (g y)) y)) singletons-are-equivalent = sol where sol : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → is-singleton X → is-singleton Y → X ≃ Y sol X Y i j = invertibility-gives-≃ (λ _ → center Y j) ((λ _ → center X i) , centrality X i , centrality Y j) NatΣ-fiber-equiv = sol where sol : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (B : X → 𝓦 ̇ ) (φ : Nat A B) (x : X) (b : B x) → fiber (φ x) b ≃ fiber (NatΣ φ) (x , b) sol A B φ x b = invertibility-gives-≃ f (g , ε , η) where f : fiber (φ x) b → fiber (NatΣ φ) (x , b) f (a , refl _) = ((x , a) , refl (x , φ x a)) g : fiber (NatΣ φ) (x , b) → fiber (φ x) b g ((x , a) , refl _) = (a , refl (φ x a)) ε : (w : fiber (φ x) b) → g (f w) ＝ w ε (a , refl _) = refl (a , refl (φ x a)) η : (t : fiber (NatΣ φ) (x , b)) → f (g t) ＝ t η ((x , a) , refl _) = refl ((x , a) , refl (NatΣ φ (x , a))) NatΣ-equiv-gives-fiberwise-equiv = sol where sol : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } (φ : Nat A B) → is-equiv (NatΣ φ) → ((x : X) → is-equiv (φ x)) sol {𝓤} {𝓥} {𝓦} {X} {A} {B} φ e x b = γ where d : fiber (φ x) b ≃ fiber (NatΣ φ) (x , b) d = NatΣ-fiber-equiv A B φ x b s : is-singleton (fiber (NatΣ φ) (x , b)) s = e (x , b) γ : is-singleton (fiber (φ x) b) γ = equiv-to-singleton d s Σ-is-subsingleton = sol where sol : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → is-subsingleton X → ((x : X) → is-subsingleton (A x)) → is-subsingleton (Σ A) sol i j (x , _) (y , _) = to-subtype-＝ j (i x y) ×-is-singleton = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-singleton X → is-singleton Y → is-singleton (X × Y) sol (x , φ) (y , γ) = (x , y) , δ where δ : ∀ z → (x , y) ＝ z δ (x' , y' ) = to-×-＝ (φ x' , γ y') ×-is-subsingleton = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-subsingleton X → is-subsingleton Y → is-subsingleton (X × Y) sol i j = Σ-is-subsingleton i (λ _ → j) ×-is-subsingleton' = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → ((Y → is-subsingleton X) × (X → is-subsingleton Y)) → is-subsingleton (X × Y) sol {𝓤} {𝓥} {X} {Y} (i , j) = k where k : is-subsingleton (X × Y) k (x , y) (x' , y') = to-×-＝ (i y x x' , j x y y') ×-is-subsingleton'-back = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-subsingleton (X × Y) → (Y → is-subsingleton X) × (X → is-subsingleton Y) sol {𝓤} {𝓥} {X} {Y} k = i , j where i : Y → is-subsingleton X i y x x' = ap pr₁ (k (x , y) (x' , y)) j : X → is-subsingleton Y j x y y' = ap pr₂ (k (x , y) (x , y')) ap₂ = sol where sol : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y → Z) {x x' : X} {y y' : Y} → x ＝ x' → y ＝ y' → f x y ＝ f x' y' sol f (refl x) (refl y) = refl (f x y)
We begin with two general results, which will be placed in a more general context later.
equiv-singleton-lemma : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (x : X) (f : (y : X) → x ＝ y → A y) → ((y : X) → is-equiv (f y)) → is-singleton (Σ A) equiv-singleton-lemma {𝓤} {𝓥} {X} {A} x f i = γ where abstract e : (y : X) → (x ＝ y) ≃ A y e y = (f y , i y) d : singleton-type' x ≃ Σ A d = Σ-cong e γ : is-singleton (Σ A) γ = equiv-to-singleton (≃-sym d) (singleton-types'-are-singletons X x) singleton-equiv-lemma : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (x : X) (f : (y : X) → x ＝ y → A y) → is-singleton (Σ A) → (y : X) → is-equiv (f y) singleton-equiv-lemma {𝓤} {𝓥} {X} {A} x f i = γ where abstract g : singleton-type' x → Σ A g = NatΣ f e : is-equiv g e = maps-of-singletons-are-equivs g (singleton-types'-are-singletons X x) i γ : (y : X) → is-equiv (f y) γ = NatΣ-equiv-gives-fiberwise-equiv f e
With this we can characterize univalence as follows:
univalence⇒ : is-univalent 𝓤 → (X : 𝓤 ̇ ) → is-singleton (Σ Y ꞉ 𝓤 ̇ , X ≃ Y) univalence⇒ ua X = equiv-singleton-lemma X (Id→Eq X) (ua X) ⇒univalence : ((X : 𝓤 ̇ ) → is-singleton (Σ Y ꞉ 𝓤 ̇ , X ≃ Y)) → is-univalent 𝓤 ⇒univalence i X = singleton-equiv-lemma X (Id→Eq X) (i X)
(Of course, this doesn’t say that there is only one type Y equivalent
to X, or only one equivalence from X to Y, because equality of
Σ types is given by transport in the second component along an
identification in the first component.)
We can replace singleton by subsingleton and still have a logical equivalence, and we sometimes need the characterization in this form:
univalence→ : is-univalent 𝓤 → (X : 𝓤 ̇ ) → is-subsingleton (Σ Y ꞉ 𝓤 ̇ , X ≃ Y) univalence→ ua X = singletons-are-subsingletons (Σ (X ≃_)) (univalence⇒ ua X) →univalence : ((X : 𝓤 ̇ ) → is-subsingleton (Σ Y ꞉ 𝓤 ̇ , X ≃ Y)) → is-univalent 𝓤 →univalence i = ⇒univalence (λ X → pointed-subsingletons-are-singletons (Σ (X ≃_)) (X , id-≃ X) (i X))
Under univalence, we get induction principles for type equivalences,
corresponding to the induction principles ℍ
and 𝕁 for identifications. To prove a
property of equivalences, it is enough to prove it for the identity
equivalence id-≃ X for all X.
Our objective is to get the induction principles ℍ-≃ and 𝕁-≃ below
and their corresponding equations. As above, it
is easy to define 𝕁-≃ from ℍ-≃, and it is harder to define ℍ-≃
directly, and it is much harder to prove the desired equation for
ℍ-≃ directly. In order to make this easy, we define an auxiliary
induction principle 𝔾-≃, from which we trivially derive ℍ-≃, and
whose equation is easy to prove.
The reason the induction principle 𝔾-≃ and its equation are easy to
construct and prove is that the type Σ Y ꞉ 𝓤 ̇ , X ≃ Y is a singleton
by univalence, which considerably simplifies reasoning about
transport. For ℍ-≃ we consider Y : 𝓤 and e : X ≃ Y separately,
whereas for 𝔾-≃ we treat them as a pair (Y , e). The point is that the
type of such pairs is a singleton by univalence.
𝔾-≃ : is-univalent 𝓤 → (X : 𝓤 ̇ ) (A : (Σ Y ꞉ 𝓤 ̇ , X ≃ Y) → 𝓥 ̇ ) → A (X , id-≃ X) → (Y : 𝓤 ̇ ) (e : X ≃ Y) → A (Y , e) 𝔾-≃ {𝓤} ua X A a Y e = transport A p a where t : Σ Y ꞉ 𝓤 ̇ , X ≃ Y t = (X , id-≃ X) p : t ＝ (Y , e) p = univalence→ {𝓤} ua X t (Y , e) 𝔾-≃-equation : (ua : is-univalent 𝓤) → (X : 𝓤 ̇ ) (A : (Σ Y ꞉ 𝓤 ̇ , X ≃ Y) → 𝓥 ̇ ) (a : A (X , id-≃ X)) → 𝔾-≃ ua X A a X (id-≃ X) ＝ a 𝔾-≃-equation {𝓤} {𝓥} ua X A a = 𝔾-≃ ua X A a X (id-≃ X) ＝⟨ refl _ ⟩ transport A p a ＝⟨ ap (λ - → transport A - a) q ⟩ transport A (refl t) a ＝⟨ refl _ ⟩ a ∎ where t : Σ Y ꞉ 𝓤 ̇ , X ≃ Y t = (X , id-≃ X) p : t ＝ t p = univalence→ {𝓤} ua X t t q : p ＝ refl t q = subsingletons-are-sets (Σ Y ꞉ 𝓤 ̇ , X ≃ Y) (univalence→ {𝓤} ua X) t t p (refl t) ℍ-≃ : is-univalent 𝓤 → (X : 𝓤 ̇ ) (A : (Y : 𝓤 ̇ ) → X ≃ Y → 𝓥 ̇ ) → A X (id-≃ X) → (Y : 𝓤 ̇ ) (e : X ≃ Y) → A Y e ℍ-≃ ua X A = 𝔾-≃ ua X (Σ-induction A) ℍ-≃-equation : (ua : is-univalent 𝓤) → (X : 𝓤 ̇ ) (A : (Y : 𝓤 ̇ ) → X ≃ Y → 𝓥 ̇ ) (a : A X (id-≃ X)) → ℍ-≃ ua X A a X (id-≃ X) ＝ a ℍ-≃-equation ua X A = 𝔾-≃-equation ua X (Σ-induction A)
The induction principle ℍ-≃ keeps X fixed and lets Y vary, while
the induction principle 𝕁-≃ lets both vary:
𝕁-≃ : is-univalent 𝓤 → (A : (X Y : 𝓤 ̇ ) → X ≃ Y → 𝓥 ̇ ) → ((X : 𝓤 ̇ ) → A X X (id-≃ X)) → (X Y : 𝓤 ̇ ) (e : X ≃ Y) → A X Y e 𝕁-≃ ua A φ X = ℍ-≃ ua X (A X) (φ X) 𝕁-≃-equation : (ua : is-univalent 𝓤) → (A : (X Y : 𝓤 ̇ ) → X ≃ Y → 𝓥 ̇ ) → (φ : (X : 𝓤 ̇ ) → A X X (id-≃ X)) → (X : 𝓤 ̇ ) → 𝕁-≃ ua A φ X X (id-≃ X) ＝ φ X 𝕁-≃-equation ua A φ X = ℍ-≃-equation ua X (A X) (φ X)
A second set of equivalence induction principles refer to is-equiv
rather than ≃ and are proved by reduction to the first version
ℍ-≃:
ℍ-equiv : is-univalent 𝓤 → (X : 𝓤 ̇ ) (A : (Y : 𝓤 ̇ ) → (X → Y) → 𝓥 ̇ ) → A X (𝑖𝑑 X) → (Y : 𝓤 ̇ ) (f : X → Y) → is-equiv f → A Y f ℍ-equiv {𝓤} {𝓥} ua X A a Y f i = γ (f , i) where B : (Y : 𝓤 ̇ ) → X ≃ Y → 𝓥 ̇ B Y (f , i) = A Y f b : B X (id-≃ X) b = a γ : (e : X ≃ Y) → B Y e γ = ℍ-≃ ua X B b Y
The above and the following say that to prove that a property of functions holds for all equivalences, it is enough to prove it for all identity functions:
𝕁-equiv : is-univalent 𝓤 → (A : (X Y : 𝓤 ̇ ) → (X → Y) → 𝓥 ̇ ) → ((X : 𝓤 ̇ ) → A X X (𝑖𝑑 X)) → (X Y : 𝓤 ̇ ) (f : X → Y) → is-equiv f → A X Y f 𝕁-equiv ua A φ X = ℍ-equiv ua X (A X) (φ X)
And the following is an immediate consequence of the fact that invertible maps are equivalences:
𝕁-invertible : is-univalent 𝓤 → (A : (X Y : 𝓤 ̇ ) → (X → Y) → 𝓥 ̇ ) → ((X : 𝓤 ̇ ) → A X X (𝑖𝑑 X)) → (X Y : 𝓤 ̇ ) (f : X → Y) → invertible f → A X Y f 𝕁-invertible ua A φ X Y f i = 𝕁-equiv ua A φ X Y f (invertibles-are-equivs f i)
For example, using ℍ-equiv we see that for any pair of functions
F : 𝓤 ̇ → 𝓤 ̇,
𝓕 : {X Y : 𝓤 ̇ } → (X → Y) → F X → F Y,
if 𝓕 preserves identities then it automatically preserves
composition of equivalences. More generally, it is enough that at least
one of the factors is an equivalence:
automatic-equiv-functoriality : (F : 𝓤 ̇ → 𝓤 ̇ ) (𝓕 : {X Y : 𝓤 ̇ } → (X → Y) → F X → F Y) (𝓕-id : {X : 𝓤 ̇ } → 𝓕 (𝑖𝑑 X) ＝ 𝑖𝑑 (F X)) {X Y Z : 𝓤 ̇ } (f : X → Y) (g : Y → Z) → is-univalent 𝓤 → is-equiv f + is-equiv g → 𝓕 (g ∘ f) ＝ 𝓕 g ∘ 𝓕 f automatic-equiv-functoriality {𝓤} F 𝓕 𝓕-id {X} {Y} {Z} f g ua = γ where γ : is-equiv f + is-equiv g → 𝓕 (g ∘ f) ＝ 𝓕 g ∘ 𝓕 f γ (inl i) = ℍ-equiv ua X A a Y f i g where A : (Y : 𝓤 ̇ ) → (X → Y) → 𝓤 ̇ A Y f = (g : Y → Z) → 𝓕 (g ∘ f) ＝ 𝓕 g ∘ 𝓕 f a : (g : X → Z) → 𝓕 g ＝ 𝓕 g ∘ 𝓕 id a g = ap (𝓕 g ∘_) (𝓕-id ⁻¹) γ (inr j) = ℍ-equiv ua Y B b Z g j f where B : (Z : 𝓤 ̇ ) → (Y → Z) → 𝓤 ̇ B Z g = (f : X → Y) → 𝓕 (g ∘ f) ＝ 𝓕 g ∘ 𝓕 f b : (f : X → Y) → 𝓕 f ＝ 𝓕 (𝑖𝑑 Y) ∘ 𝓕 f b f = ap (_∘ 𝓕 f) (𝓕-id ⁻¹)
Here is another example (see this for the terminology):
Σ-change-of-variable' : is-univalent 𝓤 → {X : 𝓤 ̇ } {Y : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (f : X → Y) → (i : is-equiv f) → (Σ x ꞉ X , A x) ＝ (Σ y ꞉ Y , A (inverse f i y)) Σ-change-of-variable' {𝓤} {𝓥} ua {X} {Y} A f i = ℍ-≃ ua X B b Y (f , i) where B : (Y : 𝓤 ̇ ) → X ≃ Y → (𝓤 ⊔ 𝓥)⁺ ̇ B Y (f , i) = Σ A ＝ (Σ (A ∘ inverse f i)) b : B X (id-≃ X) b = refl (Σ A)
An unprimed version of this is given below, after we study half adjoint equivalences.
The above version using the inverse of f can be proved directly by
induction, but the following version is perhaps more natural.
Σ-change-of-variable'' : is-univalent 𝓤 → {X : 𝓤 ̇ } {Y : 𝓤 ̇ } (A : Y → 𝓥 ̇ ) (f : X → Y) → is-equiv f → (Σ y ꞉ Y , A y) ＝ (Σ x ꞉ X , A (f x)) Σ-change-of-variable'' ua A f i = Σ-change-of-variable' ua A (inverse f i) (inverses-are-equivs f i)
This particular proof works only because inversion is involutive on the nose.
As another example we have the following:
transport-map-along-＝ : {X Y Z : 𝓤 ̇ } (p : X ＝ Y) (g : X → Z) → transport (λ - → - → Z) p g ＝ g ∘ Id→fun (p ⁻¹) transport-map-along-＝ (refl X) = refl transport-map-along-≃ : (ua : is-univalent 𝓤) {X Y Z : 𝓤 ̇ } (e : X ≃ Y) (g : X → Z) → transport (λ - → - → Z) (Eq→Id ua X Y e) g ＝ g ∘ ⌜ ≃-sym e ⌝ transport-map-along-≃ {𝓤} ua {X} {Y} {Z} = 𝕁-≃ ua A a X Y where A : (X Y : 𝓤 ̇ ) → X ≃ Y → 𝓤 ̇ A X Y e = (g : X → Z) → transport (λ - → - → Z) (Eq→Id ua X Y e) g ＝ g ∘ ⌜ ≃-sym e ⌝ a : (X : 𝓤 ̇ ) → A X X (id-≃ X) a X g = transport (λ - → - → Z) (Eq→Id ua X X (id-≃ X)) g ＝⟨ q ⟩ transport (λ - → - → Z) (refl X) g ＝⟨ refl _ ⟩ g ∎ where p : Eq→Id ua X X (id-≃ X) ＝ refl X p = inverses-are-retractions (Id→Eq X X) (ua X X) (refl X) q = ap (λ - → transport (λ - → - → Z) - g) p
An annoying feature of the use of 𝕁 (rather than pattern matching on
refl) or 𝕁-≃ is that we have to repeat what we want to prove, as
in the above examples.
An often useful alternative formulation of the notion of equivalence is that of half adjoint equivalence. If we have a function
f : X → Y
with invertibility data
g : Y → X,
η : g ∘ f ∼ id,
ε : f ∘ g ∼ id,
then for any x : X we have that
ap f (η x)
and
ε (f x)
are two identifications of type
f (g (f x)) ＝ f x.
The half adjoint condition says that these two identifications are themselves identified. The addition of the constraint
τ x : ap f (η x) ＝ ε (f x)
turns invertibility, which is data in general, into property of
f.
is-hae : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ is-hae f = Σ g ꞉ (codomain f → domain f) , Σ η ꞉ g ∘ f ∼ id , Σ ε ꞉ f ∘ g ∼ id , ((x : domain f) → ap f (η x) ＝ ε (f x))
The following just forgets the constraint τ:
haes-are-invertible : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-hae f → invertible f haes-are-invertible f (g , η , ε , τ) = g , η , ε
Hence half adjoint equivalences are equivalences, because invertible maps are equivalences. But it is also easy to prove this directly, avoiding the detour via invertible maps. We begin with a construction which will be used a number of times in connection with half adjoint equivalences.
transport-ap-≃ : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) {x x' : X} (a : x' ＝ x) (b : f x' ＝ f x) → (transport (λ - → f - ＝ f x) a b ＝ refl (f x)) ≃ (ap f a ＝ b) transport-ap-≃ f (refl x) b = γ where γ : (b ＝ refl (f x)) ≃ (refl (f x) ＝ b) γ = ⁻¹-≃ b (refl (f x)) haes-are-equivs : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-hae f → is-equiv f haes-are-equivs f (g , η , ε , τ) y = γ where c : (φ : fiber f y) → (g y , ε y) ＝ φ c (x , refl .(f x)) = q where p : transport (λ - → f - ＝ f x) (η x) (ε (f x)) ＝ refl (f x) p = ⌜ ≃-sym (transport-ap-≃ f (η x) (ε (f x))) ⌝ (τ x) q : (g (f x) , ε (f x)) ＝ (x , refl (f x)) q = to-Σ-＝ (η x , p) γ : is-singleton (fiber f y) γ = (g y , ε y) , c
To recover the constraint for all equivalences, and hence for all invertible maps, under univalence, it is enough to give the constraint for identity maps:
id-is-hae : (X : 𝓤 ̇ ) → is-hae (𝑖𝑑 X) id-is-hae X = 𝑖𝑑 X , refl , refl , (λ x → refl (refl x)) ua-equivs-are-haes : is-univalent 𝓤 → {X Y : 𝓤 ̇ } (f : X → Y) → is-equiv f → is-hae f ua-equivs-are-haes ua {X} {Y} = 𝕁-equiv ua (λ X Y f → is-hae f) id-is-hae X Y
The above can be proved without univalence as follows. This argument
also allows us to have X and Y in different universes. An example
of an equivalence of types in different universes is Id→Eq, as
stated by univalence.
equivs-are-haes : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → is-hae f equivs-are-haes {𝓤} {𝓥} {X} {Y} f i = (g , η , ε , τ) where g : Y → X g = inverse f i η : g ∘ f ∼ id η = inverses-are-retractions f i ε : f ∘ g ∼ id ε = inverses-are-sections f i τ : (x : X) → ap f (η x) ＝ ε (f x) τ x = γ where φ : fiber f (f x) φ = center (fiber f (f x)) (i (f x)) by-definition-of-g : g (f x) ＝ fiber-point φ by-definition-of-g = refl _ p : φ ＝ (x , refl (f x)) p = centrality (fiber f (f x)) (i (f x)) (x , refl (f x)) a : g (f x) ＝ x a = ap fiber-point p b : f (g (f x)) ＝ f x b = fiber-identification φ by-definition-of-η : η x ＝ a by-definition-of-η = refl _ by-definition-of-ε : ε (f x) ＝ b by-definition-of-ε = refl _ q = transport (λ - → f - ＝ f x) a b ＝⟨ refl _ ⟩ transport (λ - → f - ＝ f x) (ap pr₁ p) (pr₂ φ) ＝⟨ t ⟩ transport (λ - → f (pr₁ -) ＝ f x) p (pr₂ φ) ＝⟨ apd pr₂ p ⟩ refl (f x) ∎ where t = (transport-ap (λ - → f - ＝ f x) pr₁ p b)⁻¹ γ : ap f (η x) ＝ ε (f x) γ = ⌜ transport-ap-≃ f a b ⌝ q
We wrote the above proof of equivs-are-haes in a deliberately
verbose form to aid understanding. Here is the same proof in
a perversely reduced form:
equivs-are-haes' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → is-hae f equivs-are-haes' f e = (inverse f e , inverses-are-retractions f e , inverses-are-sections f e , τ) where τ : ∀ x → ap f (inverses-are-retractions f e x) ＝ inverses-are-sections f e (f x) τ x = ⌜ transport-ap-≃ f (ap pr₁ p) (pr₂ φ) ⌝ q where φ : fiber f (f x) φ = pr₁ (e (f x)) p : φ ＝ (x , refl (f x)) p = pr₂ (e (f x)) (x , refl (f x)) q : transport (λ - → f - ＝ f x) (ap pr₁ p) (pr₂ φ) ＝ refl (f x) q = (transport-ap (λ - → f - ＝ f x) pr₁ p ((pr₂ φ)))⁻¹ ∙ apd pr₂ p
Notice that we have the following factorization, on the nose, of the construction of invertibility data from the equivalence property:
equiv-invertible-hae-factorization : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → equivs-are-invertible f ∼ haes-are-invertible f ∘ equivs-are-haes f equiv-invertible-hae-factorization f e = refl (equivs-are-invertible f e)
Instead of working with the notion of half adjoint equivalence, we can just work with Voevodsky’s notion of equivalence, and use the fact that it satisfies the half adjoint condition:
half-adjoint-condition : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) (e : is-equiv f) (x : X) → ap f (inverses-are-retractions f e x) ＝ inverses-are-sections f e (f x) half-adjoint-condition f e = pr₂ (pr₂ (pr₂ (equivs-are-haes f e)))
Here is an example, where, compared to
Σ-change-of-variable', we
remove univalence from the hypothesis, generalize the universe of the
type Y, and weaken equality to equivalence in the conclusion. Notice
that the proof starts as that of
Σ-reindexing-retract.
Σ-change-of-variable : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (A : Y → 𝓦 ̇ ) (f : X → Y) → is-equiv f → (Σ y ꞉ Y , A y) ≃ (Σ x ꞉ X , A (f x)) Σ-change-of-variable {𝓤} {𝓥} {𝓦} {X} {Y} A f i = γ where g = inverse f i η = inverses-are-retractions f i ε = inverses-are-sections f i τ = half-adjoint-condition f i φ : Σ A → Σ (A ∘ f) φ (y , a) = (g y , transport A ((ε y)⁻¹) a) ψ : Σ (A ∘ f) → Σ A ψ (x , a) = (f x , a) ψφ : (z : Σ A) → ψ (φ z) ＝ z ψφ (y , a) = p where p : (f (g y) , transport A ((ε y)⁻¹) a) ＝ (y , a) p = to-Σ-＝ (ε y , transport-is-retraction A (ε y) a) φψ : (t : Σ (A ∘ f)) → φ (ψ t) ＝ t φψ (x , a) = p where a' : A (f (g (f x))) a' = transport A ((ε (f x))⁻¹) a q = transport (A ∘ f) (η x) a' ＝⟨ transport-ap A f (η x) a' ⟩ transport A (ap f (η x)) a' ＝⟨ ap (λ - → transport A - a') (τ x) ⟩ transport A (ε (f x)) a' ＝⟨ transport-is-retraction A (ε (f x)) a ⟩ a ∎ p : (g (f x) , transport A ((ε (f x))⁻¹) a) ＝ (x , a) p = to-Σ-＝ (η x , q) γ : Σ A ≃ Σ (A ∘ f) γ = invertibility-gives-≃ φ (ψ , ψφ , φψ)
For the sake of completeness, we also include the proof from the HoTT book that invertible maps are half adjoint equivalences, which uses a standard argument coming from category theory.
We first need some naturality lemmas:
~-naturality : {X : 𝓤 ̇ } {A : 𝓥 ̇ } (f g : X → A) (H : f ∼ g) {x y : X} {p : x ＝ y} → H x ∙ ap g p ＝ ap f p ∙ H y ~-naturality f g H {x} {_} {refl a} = refl-left ⁻¹ ~-naturality' : {X : 𝓤 ̇ } {A : 𝓥 ̇ } (f g : X → A) (H : f ∼ g) {x y : X} {p : x ＝ y} → H x ∙ ap g p ∙ (H y)⁻¹ ＝ ap f p ~-naturality' f g H {x} {x} {refl x} = ⁻¹-right∙ (H x) ~-id-naturality : {X : 𝓤 ̇ } (h : X → X) (η : h ∼ id) {x : X} → η (h x) ＝ ap h (η x) ~-id-naturality h η {x} = η (h x) ＝⟨ refl _ ⟩ η (h x) ∙ refl (h x) ＝⟨ i ⟩ η (h x) ∙ (η x ∙ (η x)⁻¹) ＝⟨ ii ⟩ η (h x) ∙ η x ∙ (η x)⁻¹ ＝⟨ iii ⟩ η (h x) ∙ ap id (η x) ∙ (η x)⁻¹ ＝⟨ iv ⟩ ap h (η x) ∎ where i = ap (η(h x) ∙_) ((⁻¹-right∙ (η x))⁻¹) ii = (∙assoc (η (h x)) (η x) (η x ⁻¹))⁻¹ iii = ap (λ - → η (h x) ∙ - ∙ η x ⁻¹) ((ap-id (η x))⁻¹) iv = ~-naturality' h id η {h x} {x} {η x}
The idea of the following proof is to improve ε to be able to give
the required τ:
invertibles-are-haes : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → invertible f → is-hae f invertibles-are-haes f (g , η , ε) = g , η , ε' , τ where ε' = λ y → f (g y) ＝⟨ (ε (f (g y)))⁻¹ ⟩ f (g (f (g y))) ＝⟨ ap f (η (g y)) ⟩ f (g y) ＝⟨ ε y ⟩ y ∎ module _ (x : domain f) where p = η (g (f x)) ＝⟨ ~-id-naturality (g ∘ f) η ⟩ ap (g ∘ f) (η x) ＝⟨ ap-∘ f g (η x) ⟩ ap g (ap f (η x)) ∎ q = ap f (η (g (f x))) ∙ ε (f x) ＝⟨ by-p ⟩ ap f (ap g (ap f (η x))) ∙ ε (f x) ＝⟨ by-ap-∘ ⟩ ap (f ∘ g) (ap f (η x)) ∙ ε (f x) ＝⟨ by-~-naturality ⟩ ε (f (g (f x))) ∙ ap id (ap f (η x)) ＝⟨ by-ap-id ⟩ ε (f (g (f x))) ∙ ap f (η x) ∎ where by-p = ap (λ - → ap f - ∙ ε (f x)) p by-ap-∘ = ap (_∙ ε (f x)) ((ap-∘ g f (ap f (η x)))⁻¹) by-~-naturality = (~-naturality (f ∘ g) id ε {f (g (f x))} {f x} {ap f (η x)})⁻¹ by-ap-id = ap (ε (f (g (f x))) ∙_) (ap-id (ap f (η x))) τ = ap f (η x) ＝⟨ refl-left ⁻¹ ⟩ refl (f (g (f x))) ∙ ap f (η x) ＝⟨ by-⁻¹-left∙ ⟩ (ε (f (g (f x))))⁻¹ ∙ ε (f (g (f x))) ∙ ap f (η x) ＝⟨ by-∙assoc ⟩ (ε (f (g (f x))))⁻¹ ∙ (ε (f (g (f x))) ∙ ap f (η x)) ＝⟨ by-q ⟩ (ε (f (g (f x))))⁻¹ ∙ (ap f (η (g (f x))) ∙ ε (f x)) ＝⟨ refl _ ⟩ ε' (f x) ∎ where by-⁻¹-left∙ = ap (_∙ ap f (η x)) ((⁻¹-left∙ (ε (f (g (f x)))))⁻¹) by-∙assoc = ∙assoc ((ε (f (g (f x))))⁻¹) (ε (f (g (f x)))) (ap f (η x)) by-q = ap ((ε (f (g (f x))))⁻¹ ∙_) (q ⁻¹)
Function extensionality says that any two pointwise equal functions
are equal. This is known to be not provable or disprovable in
MLTT. It is an independent statement, which we abbreviate as funext.
funext : ∀ 𝓤 𝓥 → (𝓤 ⊔ 𝓥)⁺ ̇ funext 𝓤 𝓥 = {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {f g : X → Y} → f ∼ g → f ＝ g
There will be two seemingly stronger
statements, namely the generalization to dependent functions, and the
requirement that the canonical map f ＝ g → f ∼ g is an
equivalence.
Exercise. Assuming funext, prove that if a function f : X → Y is
an equivalence then so is the precomposition map _∘ f : (Y → Z) →
(X → Z).
The crucial step in Voevodsky’s
proof
that univalence implies funext is to establish the conclusion of the
above exercise assuming univalence instead. We prove this by
equivalence induction on
f, which means that we only need to consider the case when f is an
identity function, for which precomposition with f is itself an
identity function (of a function type), and hence an equivalence:
precomp-is-equiv : is-univalent 𝓤 → (X Y : 𝓤 ̇ ) (f : X → Y) → is-equiv f → (Z : 𝓤 ̇ ) → is-equiv (λ (g : Y → Z) → g ∘ f) precomp-is-equiv {𝓤} ua = 𝕁-equiv ua (λ X Y (f : X → Y) → (Z : 𝓤 ̇ ) → is-equiv (λ g → g ∘ f)) (λ X Z → id-is-equiv (X → Z))
With this we can prove the desired result as follows.
univalence-gives-funext : is-univalent 𝓤 → funext 𝓥 𝓤 univalence-gives-funext {𝓤} {𝓥} ua {X} {Y} {f₀} {f₁} = γ where Δ : 𝓤 ̇ Δ = Σ y₀ ꞉ Y , Σ y₁ ꞉ Y , y₀ ＝ y₁ δ : Y → Δ δ y = (y , y , refl y) π₀ π₁ : Δ → Y π₀ (y₀ , y₁ , p) = y₀ π₁ (y₀ , y₁ , p) = y₁ δ-is-equiv : is-equiv δ δ-is-equiv = invertibles-are-equivs δ (π₀ , η , ε) where η : (y : Y) → π₀ (δ y) ＝ y η y = refl y ε : (d : Δ) → δ (π₀ d) ＝ d ε (y , y , refl y) = refl (y , y , refl y) φ : (Δ → Y) → (Y → Y) φ π = π ∘ δ φ-is-equiv : is-equiv φ φ-is-equiv = precomp-is-equiv ua Y Δ δ δ-is-equiv Y p : φ π₀ ＝ φ π₁ p = refl (𝑖𝑑 Y) q : π₀ ＝ π₁ q = equivs-are-lc φ φ-is-equiv p γ : f₀ ∼ f₁ → f₀ ＝ f₁ γ h = ap (λ π x → π (f₀ x , f₁ x , h x)) q
This definition of γ is probably too concise. Here are all the steps
performed silently by Agda, by expanding judgmental equalities,
indicated with refl here:
γ' : f₀ ∼ f₁ → f₀ ＝ f₁ γ' h = f₀ ＝⟨ refl _ ⟩ (λ x → f₀ x) ＝⟨ refl _ ⟩ (λ x → π₀ (f₀ x , f₁ x , h x)) ＝⟨ ap (λ - x → - (f₀ x , f₁ x , h x)) q ⟩ (λ x → π₁ (f₀ x , f₁ x , h x)) ＝⟨ refl _ ⟩ (λ x → f₁ x) ＝⟨ refl _ ⟩ f₁ ∎
So notice that this relies on the so-called η-rule for judgmental equality of functions, namely
f = λ x → f x.
Without it, we would only get that
f₀ ∼ f₁ → (λ x → f₀ x) ＝ (λ x → f₁ x)
instead.
Dependent function extensionality:
dfunext : ∀ 𝓤 𝓥 → (𝓤 ⊔ 𝓥)⁺ ̇ dfunext 𝓤 𝓥 = {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {f g : Π A} → f ∼ g → f ＝ g
The above says that there is some map f ~ g → f ＝ g. The following
instead says that the canonical map happly in the other direction is
an equivalence:
happly : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (f g : Π A) → f ＝ g → f ∼ g happly f g p x = ap (λ - → - x) p hfunext : ∀ 𝓤 𝓥 → (𝓤 ⊔ 𝓥)⁺ ̇ hfunext 𝓤 𝓥 = {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (f g : Π A) → is-equiv (happly f g) hfunext-gives-dfunext : hfunext 𝓤 𝓥 → dfunext 𝓤 𝓥 hfunext-gives-dfunext hfe {X} {A} {f} {g} = inverse (happly f g) (hfe f g)
Voevodsky showed that all these notions of function extensionality are logically equivalent to saying that products of singletons are singletons:
vvfunext : ∀ 𝓤 𝓥 → (𝓤 ⊔ 𝓥)⁺ ̇ vvfunext 𝓤 𝓥 = {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-singleton (A x)) → is-singleton (Π A) dfunext-gives-vvfunext : dfunext 𝓤 𝓥 → vvfunext 𝓤 𝓥 dfunext-gives-vvfunext fe {X} {A} i = γ where f : Π A f x = center (A x) (i x) c : (g : Π A) → f ＝ g c g = fe (λ (x : X) → centrality (A x) (i x) (g x)) γ : is-singleton (Π A) γ = f , c vvfunext-gives-hfunext : vvfunext 𝓤 𝓥 → hfunext 𝓤 𝓥 vvfunext-gives-hfunext vfe {X} {Y} f = γ where a : (x : X) → is-singleton (Σ y ꞉ Y x , f x ＝ y) a x = singleton-types'-are-singletons (Y x) (f x) c : is-singleton (Π x ꞉ X , Σ y ꞉ Y x , f x ＝ y) c = vfe a ρ : (Σ g ꞉ Π Y , f ∼ g) ◁ (Π x ꞉ X , Σ y ꞉ Y x , f x ＝ y) ρ = ≃-gives-▷ ΠΣ-distr-≃ d : is-singleton (Σ g ꞉ Π Y , f ∼ g) d = retract-of-singleton ρ c e : (Σ g ꞉ Π Y , f ＝ g) → (Σ g ꞉ Π Y , f ∼ g) e = NatΣ (happly f) i : is-equiv e i = maps-of-singletons-are-equivs e (singleton-types'-are-singletons (Π Y) f) d γ : (g : Π Y) → is-equiv (happly f g) γ = NatΣ-equiv-gives-fiberwise-equiv (happly f) i
And finally the seemingly rather weak, non-dependent version funext
implies the seemingly strongest version, which closes the circle of
logical equivalences. We first need some lemmas.
postcomp-invertible : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : 𝓦 ̇ } → funext 𝓦 𝓤 → funext 𝓦 𝓥 → (f : X → Y) → invertible f → invertible (λ (h : A → X) → f ∘ h) postcomp-invertible {𝓤} {𝓥} {𝓦} {X} {Y} {A} fe fe' f (g , η , ε) = γ where f' : (A → X) → (A → Y) f' h = f ∘ h g' : (A → Y) → (A → X) g' k = g ∘ k η' : (h : A → X) → g' (f' h) ＝ h η' h = fe (η ∘ h) ε' : (k : A → Y) → f' (g' k) ＝ k ε' k = fe' (ε ∘ k) γ : invertible f' γ = (g' , η' , ε') postcomp-is-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : 𝓦 ̇ } → funext 𝓦 𝓤 → funext 𝓦 𝓥 → (f : X → Y) → is-equiv f → is-equiv (λ (h : A → X) → f ∘ h) postcomp-is-equiv fe fe' f e = invertibles-are-equivs (λ h → f ∘ h) (postcomp-invertible fe fe' f (equivs-are-invertible f e)) funext-gives-vvfunext : funext 𝓤 (𝓤 ⊔ 𝓥) → funext 𝓤 𝓤 → vvfunext 𝓤 𝓥 funext-gives-vvfunext {𝓤} {𝓥} fe fe' {X} {A} φ = γ where f : Σ A → X f = pr₁ f-is-equiv : is-equiv f f-is-equiv = pr₁-is-equiv φ g : (X → Σ A) → (X → X) g h = f ∘ h e : is-equiv g e = postcomp-is-equiv fe fe' f f-is-equiv i : is-singleton (Σ h ꞉ (X → Σ A), f ∘ h ＝ 𝑖𝑑 X) i = e (𝑖𝑑 X) r : (Σ h ꞉ (X → Σ A), f ∘ h ＝ 𝑖𝑑 X) → Π A r (h , p) x = transport A (happly (f ∘ h) (𝑖𝑑 X) p x) (pr₂ (h x)) s : Π A → (Σ h ꞉ (X → Σ A), f ∘ h ＝ 𝑖𝑑 X) s ψ = (λ x → x , ψ x) , refl (𝑖𝑑 X) η : ∀ ψ → r (s ψ) ＝ ψ η ψ = refl (r (s ψ)) γ : is-singleton (Π A) γ = retract-of-singleton (r , s , η) i
We have the following corollaries. We first formulate the types of some functions:
abstract funext-gives-hfunext : funext 𝓤 (𝓤 ⊔ 𝓥) → funext 𝓤 𝓤 → hfunext 𝓤 𝓥 dfunext-gives-hfunext : dfunext 𝓤 𝓥 → hfunext 𝓤 𝓥 funext-gives-dfunext : funext 𝓤 (𝓤 ⊔ 𝓥) → funext 𝓤 𝓤 → dfunext 𝓤 𝓥 univalence-gives-dfunext' : is-univalent 𝓤 → is-univalent (𝓤 ⊔ 𝓥) → dfunext 𝓤 𝓥 univalence-gives-hfunext' : is-univalent 𝓤 → is-univalent (𝓤 ⊔ 𝓥) → hfunext 𝓤 𝓥 univalence-gives-vvfunext' : is-univalent 𝓤 → is-univalent (𝓤 ⊔ 𝓥) → vvfunext 𝓤 𝓥 univalence-gives-hfunext : is-univalent 𝓤 → hfunext 𝓤 𝓤 univalence-gives-dfunext : is-univalent 𝓤 → dfunext 𝓤 𝓤 univalence-gives-vvfunext : is-univalent 𝓤 → vvfunext 𝓤 𝓤
And then we give their definitions (Agda makes sure there are no circularities):
funext-gives-hfunext fe fe' = vvfunext-gives-hfunext (funext-gives-vvfunext fe fe') funext-gives-dfunext fe fe' = hfunext-gives-dfunext (funext-gives-hfunext fe fe') dfunext-gives-hfunext fe = vvfunext-gives-hfunext (dfunext-gives-vvfunext fe) univalence-gives-dfunext' ua ua' = funext-gives-dfunext (univalence-gives-funext ua') (univalence-gives-funext ua) univalence-gives-hfunext' ua ua' = funext-gives-hfunext (univalence-gives-funext ua') (univalence-gives-funext ua) univalence-gives-vvfunext' ua ua' = funext-gives-vvfunext (univalence-gives-funext ua') (univalence-gives-funext ua) univalence-gives-hfunext ua = univalence-gives-hfunext' ua ua univalence-gives-dfunext ua = univalence-gives-dfunext' ua ua univalence-gives-vvfunext ua = univalence-gives-vvfunext' ua ua
Under univalence, a universe 𝓤 becomes a map classifier, in the
sense that maps from a type X in 𝓤 into a type Y in 𝓤 are in
canonical bijection with functions Y → 𝓤. Using the following
slice notation, this amounts to a bijection between 𝓤 / Y and Y → 𝓤:
_/_ : (𝓤 : Universe) → 𝓥 ̇ → 𝓤 ⁺ ⊔ 𝓥 ̇ 𝓤 / Y = Σ X ꞉ 𝓤 ̇ , (X → Y)
We need the following lemma, which has other uses:
total-fiber-is-domain : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → Σ (fiber f) ≃ X total-fiber-is-domain {𝓤} {𝓥} {X} {Y} f = invertibility-gives-≃ g (h , η , ε) where g : (Σ y ꞉ Y , Σ x ꞉ X , f x ＝ y) → X g (y , x , p) = x h : X → Σ y ꞉ Y , Σ x ꞉ X , f x ＝ y h x = (f x , x , refl (f x)) η : ∀ t → h (g t) ＝ t η (_ , x , refl _) = refl (f x , x , refl _) ε : (x : X) → g (h x) ＝ x ε = refl
The function χ gives the characteristic function of a map into Y:
χ : (Y : 𝓤 ̇ ) → 𝓤 / Y → (Y → 𝓤 ̇ ) χ Y (X , f) = fiber f
We say that a universe is a map classifier if the above function is an equivalence for every Y in the universe:
is-map-classifier : (𝓤 : Universe) → 𝓤 ⁺ ̇ is-map-classifier 𝓤 = (Y : 𝓤 ̇ ) → is-equiv (χ Y)
Any Y → 𝓤 is the characteristic function of some map into Y by
taking its total space and the first projection:
𝕋 : (Y : 𝓤 ̇ ) → (Y → 𝓤 ̇ ) → 𝓤 / Y 𝕋 Y A = Σ A , pr₁ χη : is-univalent 𝓤 → (Y : 𝓤 ̇ ) (σ : 𝓤 / Y) → 𝕋 Y (χ Y σ) ＝ σ χη ua Y (X , f) = r where e : Σ (fiber f) ≃ X e = total-fiber-is-domain f p : Σ (fiber f) ＝ X p = Eq→Id ua (Σ (fiber f)) X e observation : ⌜ ≃-sym e ⌝ ＝ (λ x → f x , x , refl (f x)) observation = refl _ q = transport (λ - → - → Y) p pr₁ ＝⟨ transport-map-along-≃ ua e pr₁ ⟩ pr₁ ∘ ⌜ ≃-sym e ⌝ ＝⟨ refl _ ⟩ f ∎ r : (Σ (fiber f) , pr₁) ＝ (X , f) r = to-Σ-＝ (p , q) χε : is-univalent 𝓤 → dfunext 𝓤 (𝓤 ⁺) → (Y : 𝓤 ̇ ) (A : Y → 𝓤 ̇ ) → χ Y (𝕋 Y A) ＝ A χε ua fe Y A = fe γ where f : ∀ y → fiber pr₁ y → A y f y ((y , a) , refl p) = a g : ∀ y → A y → fiber pr₁ y g y a = (y , a) , refl y η : ∀ y σ → g y (f y σ) ＝ σ η y ((y , a) , refl p) = refl ((y , a) , refl p) ε : ∀ y a → f y (g y a) ＝ a ε y a = refl a γ : ∀ y → fiber pr₁ y ＝ A y γ y = Eq→Id ua _ _ (invertibility-gives-≃ (f y) (g y , η y , ε y)) universes-are-map-classifiers : is-univalent 𝓤 → dfunext 𝓤 (𝓤 ⁺) → is-map-classifier 𝓤 universes-are-map-classifiers ua fe Y = invertibles-are-equivs (χ Y) (𝕋 Y , χη ua Y , χε ua fe Y)
Therefore we have the following canonical equivalence:
map-classification : is-univalent 𝓤 → dfunext 𝓤 (𝓤 ⁺) → (Y : 𝓤 ̇ ) → 𝓤 / Y ≃ (Y → 𝓤 ̇ ) map-classification ua fe Y = χ Y , universes-are-map-classifiers ua fe Y
If we use a type as an axiom, it should better have at most one element. We prove some generally useful lemmas first.
Π-is-subsingleton : dfunext 𝓤 𝓥 → {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-subsingleton (A x)) → is-subsingleton (Π A) Π-is-subsingleton fe i f g = fe (λ x → i x (f x) (g x)) being-singleton-is-subsingleton : dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } → is-subsingleton (is-singleton X) being-singleton-is-subsingleton fe {X} (x , φ) (y , γ) = p where i : is-subsingleton X i = singletons-are-subsingletons X (y , γ) s : is-set X s = subsingletons-are-sets X i a : (z : X) → is-subsingleton ((t : X) → z ＝ t) a z = Π-is-subsingleton fe (s z) b : x ＝ y b = φ y p : (x , φ) ＝ (y , γ) p = to-subtype-＝ a b being-equiv-is-subsingleton : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-subsingleton (is-equiv f) being-equiv-is-subsingleton fe fe' f = Π-is-subsingleton fe (λ x → being-singleton-is-subsingleton fe')
In passing, we fulfill a promise made above:
subsingletons-are-retracts-of-logically-equivalent-types : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-subsingleton X → (X ⇔ Y) → X ◁ Y subsingletons-are-retracts-of-logically-equivalent-types i (f , g) = g , f , η where η : g ∘ f ∼ id η x = i (g (f x)) x equivalence-property-is-retract-of-invertibility-data : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f ◁ invertible f equivalence-property-is-retract-of-invertibility-data fe fe' f = subsingletons-are-retracts-of-logically-equivalent-types (being-equiv-is-subsingleton fe fe' f) (equivs-are-invertible f , invertibles-are-equivs f)
We now return to our main concern in this section.
univalence-is-subsingleton : is-univalent (𝓤 ⁺) → is-subsingleton (is-univalent 𝓤) univalence-is-subsingleton {𝓤} ua⁺ = subsingleton-criterion' γ where γ : is-univalent 𝓤 → is-subsingleton (is-univalent 𝓤) γ ua = i where dfe₁ : dfunext 𝓤 (𝓤 ⁺) dfe₂ : dfunext (𝓤 ⁺) (𝓤 ⁺) dfe₁ = univalence-gives-dfunext' ua ua⁺ dfe₂ = univalence-gives-dfunext ua⁺ i : is-subsingleton (is-univalent 𝓤) i = Π-is-subsingleton dfe₂ (λ X → Π-is-subsingleton dfe₂ (λ Y → being-equiv-is-subsingleton dfe₁ dfe₂ (Id→Eq X Y)))
So if all universes are univalent then “being univalent” is a
subsingleton, and hence a singleton. This hypothesis of global
univalence cannot be expressed in our MLTT that only has ω many
universes, because global univalence would have to live in the first
universe after them. Agda does
have
such a universe 𝓤ω, and so we can formulate it here. There would be
no problem in extending our MLTT to have such a universe if we so
wished, in which case we would be able to formulate and prove:
Univalence : 𝓤ω Univalence = ∀ 𝓤 → is-univalent 𝓤 univalence-is-subsingletonω : Univalence → is-subsingleton (is-univalent 𝓤) univalence-is-subsingletonω {𝓤} γ = univalence-is-subsingleton (γ (𝓤 ⁺)) univalence-is-a-singleton : Univalence → is-singleton (is-univalent 𝓤) univalence-is-a-singleton {𝓤} γ = pointed-subsingletons-are-singletons (is-univalent 𝓤) (γ 𝓤) (univalence-is-subsingletonω γ)
That the type Univalence would be a subsingleton can’t even be
formulated in the absence of a successor 𝓤ω⁺ of 𝓤ω, and Agda
doesn’t have such a successor universe (but there isn’t any
fundamental reason why it couldn’t have it).
In the absence of a universe 𝓤ω in our MLTT, we can simply have an
axiom schema, consisting
of ω-many axioms, stating that each universe is univalent. Then we
can prove in our MLTT that the univalence property for each universe is
a (sub)singleton, with ω-many proofs (or just one schematic proof
with a free variable for a universe 𝓤ₙ).
It follows immediately from the above that global univalence gives global function extensionality.
global-dfunext : 𝓤ω global-dfunext = ∀ {𝓤 𝓥} → dfunext 𝓤 𝓥 univalence-gives-global-dfunext : Univalence → global-dfunext univalence-gives-global-dfunext ua {𝓤} {𝓥} = univalence-gives-dfunext' (ua 𝓤) (ua (𝓤 ⊔ 𝓥)) global-hfunext : 𝓤ω global-hfunext = ∀ {𝓤 𝓥} → hfunext 𝓤 𝓥 univalence-gives-global-hfunext : Univalence → global-hfunext univalence-gives-global-hfunext ua {𝓤} {𝓥} = univalence-gives-hfunext' (ua 𝓤) (ua (𝓤 ⊔ 𝓥))
vvfunext 𝓤 𝓥 and hfunext 𝓤 𝓥 are (sub)singletonsWe need a version of
Π-is-subsingleton for
dependent functions with implicit arguments.
Π-is-subsingleton' : dfunext 𝓤 𝓥 → {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-subsingleton (A x)) → is-subsingleton ({x : X} → A x) Π-is-subsingleton' fe {X} {A} i = γ where ρ : ({x : X} → A x) ◁ Π A ρ = (λ f {x} → f x) , (λ g x → g {x}) , refl γ : is-subsingleton ({x : X} → A x) γ = retract-of-subsingleton ρ (Π-is-subsingleton fe i)
To show that vvfunext 𝓤 𝓥 and hfunext 𝓤 𝓥 are subsingletons, we
need assumptions of function extensionality for higher universes:
vv-and-hfunext-are-subsingletons-lemma : dfunext (𝓤 ⁺) (𝓤 ⊔ (𝓥 ⁺)) → dfunext (𝓤 ⊔ (𝓥 ⁺)) (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → is-subsingleton (vvfunext 𝓤 𝓥) × is-subsingleton (hfunext 𝓤 𝓥) vv-and-hfunext-are-subsingletons-lemma {𝓤} {𝓥} dfe dfe' dfe'' = φ , γ where φ : is-subsingleton (vvfunext 𝓤 𝓥) φ = Π-is-subsingleton' dfe (λ X → Π-is-subsingleton' dfe' (λ A → Π-is-subsingleton dfe'' (λ i → being-singleton-is-subsingleton dfe''))) γ : is-subsingleton (hfunext 𝓤 𝓥) γ = Π-is-subsingleton' dfe (λ X → Π-is-subsingleton' dfe' (λ A → Π-is-subsingleton dfe'' (λ f → Π-is-subsingleton dfe'' (λ g → being-equiv-is-subsingleton dfe'' dfe'' (happly f g)))))
Hence they are singletons assuming global univalence (or just global function extensionality, of any kind):
vv-and-hfunext-are-singletons : Univalence → is-singleton (vvfunext 𝓤 𝓥) × is-singleton (hfunext 𝓤 𝓥) vv-and-hfunext-are-singletons {𝓤} {𝓥} ua = f (vv-and-hfunext-are-subsingletons-lemma (univalence-gives-dfunext' (ua (𝓤 ⁺)) (ua ((𝓤 ⁺) ⊔ (𝓥 ⁺)))) (univalence-gives-dfunext' (ua (𝓤 ⊔ (𝓥 ⁺))) (ua (𝓤 ⊔ (𝓥 ⁺)))) (univalence-gives-dfunext' (ua (𝓤 ⊔ 𝓥)) (ua (𝓤 ⊔ 𝓥)))) where f : is-subsingleton (vvfunext 𝓤 𝓥) × is-subsingleton (hfunext 𝓤 𝓥) → is-singleton (vvfunext 𝓤 𝓥) × is-singleton (hfunext 𝓤 𝓥) f (i , j) = pointed-subsingletons-are-singletons (vvfunext 𝓤 𝓥) (univalence-gives-vvfunext' (ua 𝓤) (ua (𝓤 ⊔ 𝓥))) i , pointed-subsingletons-are-singletons (hfunext 𝓤 𝓥) (univalence-gives-hfunext' (ua 𝓤) (ua (𝓤 ⊔ 𝓥))) j
However, funext 𝓤 𝓤 and dfunext 𝓤 𝓤 are not subsingletons (see the
HoTT book).
Unique existence of x : X with A x in univalent mathematics, written
∃! x ꞉ X , A x
or simply
∃! A,
requires that not only the point x : X but also the datum a : A x to be
unique. More precisely, we require that there is a unique pair (x ,
a) : Σ A.
This is particularly important in the formulation of universal properties involving types that are not necessarily sets, where it generalizes the categorical notion of uniqueness up to unique isomorphism.
∃! : {X : 𝓤 ̇ } → (X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ ∃! A = is-singleton (Σ A) -∃! : {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) (Y : X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ -∃! X Y = ∃! Y syntax -∃! A (λ x → b) = ∃! x ꞉ A , b ∃!-is-subsingleton : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → is-subsingleton (∃! A) ∃!-is-subsingleton A fe = being-singleton-is-subsingleton fe unique-existence-gives-weak-unique-existence : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → (∃! x ꞉ X , A x) → (Σ x ꞉ X , A x) × ((x y : X) → A x → A y → x ＝ y) unique-existence-gives-weak-unique-existence A s = center (Σ A) s , u where u : ∀ x y → A x → A y → x ＝ y u x y a b = ap pr₁ (singletons-are-subsingletons (Σ A) s (x , a) (y , b))
The converse holds if each A x is a subsingleton:
weak-unique-existence-gives-unique-existence-sometimes : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → ((x : X) → is-subsingleton (A x)) → ((Σ x ꞉ X , A x) × ((x y : X) → A x → A y → x ＝ y)) → (∃! x ꞉ X , A x) weak-unique-existence-gives-unique-existence-sometimes A i ((x , a) , u) = (x , a) , φ where φ : (σ : Σ A) → x , a ＝ σ φ (y , b) = to-subtype-＝ i (u x y a b)
Exercise. Find a counter-example in the absence of the requirement
that all types A x are subsingletons.
Similarly, the existence of at most one x : X with A x should be
understood as
is-subsingleton (Σ A),
but we will not introduce special notation for this concept, although it will occur often.
The natural numbers have the following universal
property. What is
noteworthy here is that the type Y need not be a set, so that the
two equations can hold in multiple ways, but nevertheless we have
unique existence in the sense of the previous section. Moreover,
univalence is not needed. Function extensionality suffices.
ℕ-is-nno : hfunext 𝓤₀ 𝓤 → (Y : 𝓤 ̇ ) (y₀ : Y) (g : Y → Y) → ∃! h ꞉ (ℕ → Y), (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h) ℕ-is-nno {𝓤} hfe Y y₀ g = γ where
We apply the same retraction techniques we used in order to prove that
invertible maps are
equivalences. We first show that, for any h : ℕ → Y, the type
(h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h)
is a retract of the type
h ∼ ℕ-iteration Y y₀ g
and hence, by function extensionality, we also have a retraction if we
replace pointwise equality ∼ by equality ＝. Thus the type
Σ h ꞉ ℕ → Y , (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h)
is a retract of the singleton type
Σ h ꞉ ℕ → Y , h ＝ ℕ-iteration Y y₀ g,
and therefore is itself a singleton, as required.
Now we do this in Agda. We first establish the following retraction (which is automatically an equivalence, but we don’t need this fact).
lemma₀ : (h : ℕ → Y) → ((h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h)) ◁ (h ∼ ℕ-iteration Y y₀ g) lemma₀ h = r , s , η where s : (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h) → h ∼ ℕ-iteration Y y₀ g s (p , K) 0 = p s (p , K) (succ n) = h (succ n) ＝⟨ K n ⟩ g (h n) ＝⟨ ap g (s (p , K) n) ⟩ ℕ-iteration Y y₀ g (succ n) ∎
The above section s is defined by induction on natural numbers, but
the following retraction r is defined directly.
r : codomain s → domain s
r H = H 0 , (λ n → h (succ n) ＝⟨ H (succ n) ⟩
g (ℕ-iteration Y y₀ g n) ＝⟨ ap g ((H n)⁻¹) ⟩
g (h n ) ∎)
The retraction property doesn’t need induction on natural numbers:
η : (z : (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h)) → r (s z) ＝ z
η (p , K) = v
where
i = λ n →
K n ∙ ap g (s (p , K) n) ∙ ap g ((s (p , K) n) ⁻¹) ＝⟨ ii n ⟩
K n ∙ (ap g (s (p , K) n) ∙ ap g ((s (p , K) n) ⁻¹)) ＝⟨ iii n ⟩
K n ∙ (ap g (s (p , K) n) ∙ (ap g (s (p , K) n))⁻¹) ＝⟨ iv n ⟩
K n ∎
where
ii = λ n → ∙assoc (K n) (ap g (s (p , K) n)) (ap g ((s (p , K) n)⁻¹))
iii = λ n → ap (λ - → K n ∙ (ap g (s (p , K) n) ∙ -)) (ap⁻¹ g (s (p , K) n) ⁻¹)
iv = λ n → ap (K n ∙_) (⁻¹-right∙ (ap g (s (p , K) n)))
v : (p , (λ n → s (p , K) (succ n) ∙ ap g ((s (p , K) n)⁻¹))) ＝ (p , K)
v = ap (p ,_) (hfunext-gives-dfunext hfe i)
lemma₁ = λ h → (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h) ◁⟨ lemma₀ h ⟩
(h ∼ ℕ-iteration Y y₀ g) ◁⟨ i h ⟩
(h ＝ ℕ-iteration Y y₀ g) ◀
where
i = λ h → ≃-gives-▷ (happly h (ℕ-iteration Y y₀ g) , hfe _ _)
lemma₂ : (Σ h ꞉ (ℕ → Y), (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h))
◁ (Σ h ꞉ (ℕ → Y), h ＝ ℕ-iteration Y y₀ g)
lemma₂ = Σ-retract lemma₁
γ : is-singleton (Σ h ꞉ (ℕ → Y), (h 0 ＝ y₀) × (h ∘ succ ∼ g ∘ h))
γ = retract-of-singleton
lemma₂
(singleton-types-are-singletons (ℕ → Y) (ℕ-iteration Y y₀ g))
This concludes the proof of ℕ-is-nno. We say that ℕ is a natural
numbers object,
or, more precisely, the triple (ℕ , 0 , succ) is a natural numbers
object.
Here is an important example, which given any n : ℕ constructs a
type with n elements (and decidable equality):
module finite-types (hfe : hfunext 𝓤₀ 𝓤₁) where fin : ∃! Fin ꞉ (ℕ → 𝓤₀ ̇ ) , (Fin 0 ＝ 𝟘) × ((n : ℕ) → Fin (succ n) ＝ Fin n + 𝟙) fin = ℕ-is-nno hfe (𝓤₀ ̇ ) 𝟘 (_+ 𝟙) Fin : ℕ → 𝓤₀ ̇ Fin = pr₁ (center _ fin)
We could use the other projections to derive the following two equations, but they hold definitionally:
Fin-equation₀ : Fin 0 ＝ 𝟘 Fin-equation₀ = refl _ Fin-equation-succ : Fin ∘ succ ＝ λ n → Fin n + 𝟙 Fin-equation-succ = refl _
We also have
Fin-equation-succ' : (n : ℕ) → Fin (succ n) ＝ Fin n + 𝟙 Fin-equation-succ' n = refl _
and the examples
Fin-equation₁ : Fin 1 ＝ 𝟘 + 𝟙 Fin-equation₁ = refl _ Fin-equation₂ : Fin 2 ＝ (𝟘 + 𝟙) + 𝟙 Fin-equation₂ = refl _ Fin-equation₃ : Fin 3 ＝ ((𝟘 + 𝟙) + 𝟙) + 𝟙 Fin-equation₃ = refl _
Exercise. Assume univalence. The equation
Fin (succ n) ＝ Fin n + 𝟙
holds in multiple ways, because Fin n has n!
automorphisms. Construct an involutive fiberwise equivalence
mirror : (n : ℕ) → Fin n → Fin n
different from the identity and hence an identification Fin ＝ Fin
different from refl Fin. Consider
∃! Fin' ꞉ ℕ → 𝓤₀ ̇ , (Fin' 0 ＝ 𝟘) × (Fin' ∘ succ ＝ λ n → 𝟙 + Fin' n)
and show that Fin' (succ n) ＝ Fin' n + 𝟙 so that Fin'
satisfies the defining equations of Fin, although not judgmentally,
and hence Fin' ＝ Fin by the univeral property of Fin. But there are
many equalities Fin' n ＝ Fin n. Which one do we get by the universal
property? Show that the type Fin n has decidable equality and hence
is a set.
being-subsingleton-is-subsingleton : dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } → is-subsingleton (is-subsingleton X) being-subsingleton-is-subsingleton fe {X} i j = c where l : is-set X l = subsingletons-are-sets X i a : (x y : X) → i x y ＝ j x y a x y = l x y (i x y) (j x y) b : (x : X) → i x ＝ j x b x = fe (a x) c : i ＝ j c = fe b being-center-is-subsingleton : dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } (c : X) → is-subsingleton (is-center X c) being-center-is-subsingleton fe {X} c φ γ = k where i : is-singleton X i = c , φ j : (x : X) → is-subsingleton (c ＝ x) j x = singletons-are-sets X i c x k : φ ＝ γ k = fe (λ x → j x (φ x) (γ x))
Here the version hfunext of function extensionality is what is
needed:
Π-is-set : hfunext 𝓤 𝓥 → {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-set (A x)) → is-set (Π A) Π-is-set hfe s f g = b where a : is-subsingleton (f ∼ g) a p q = γ where h : ∀ x → p x ＝ q x h x = s x (f x) (g x) (p x) (q x) γ : p ＝ q γ = hfunext-gives-dfunext hfe h e : (f ＝ g) ≃ (f ∼ g) e = (happly f g , hfe f g) b : is-subsingleton (f ＝ g) b = equiv-to-subsingleton e a being-set-is-subsingleton : dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } → is-subsingleton (is-set X) being-set-is-subsingleton fe = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → being-subsingleton-is-subsingleton fe))
More generally:
hlevel-relation-is-subsingleton : dfunext 𝓤 𝓤 → (n : ℕ) (X : 𝓤 ̇ ) → is-subsingleton (X is-of-hlevel n) hlevel-relation-is-subsingleton {𝓤} fe zero X = being-singleton-is-subsingleton fe hlevel-relation-is-subsingleton fe (succ n) X = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ x' → hlevel-relation-is-subsingleton fe n (x ＝ x')))
Composition of equivalences is associative:
●-assoc : dfunext 𝓣 (𝓤 ⊔ 𝓣) → dfunext (𝓤 ⊔ 𝓣) (𝓤 ⊔ 𝓣) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } {T : 𝓣 ̇ } (α : X ≃ Y) (β : Y ≃ Z) (γ : Z ≃ T) → α ● (β ● γ) ＝ (α ● β) ● γ ●-assoc fe fe' (f , a) (g , b) (h , c) = ap (h ∘ g ∘ f ,_) q where d e : is-equiv (h ∘ g ∘ f) d = ∘-is-equiv (∘-is-equiv c b) a e = ∘-is-equiv c (∘-is-equiv b a) q : d ＝ e q = being-equiv-is-subsingleton fe fe' (h ∘ g ∘ f) _ _ ≃-sym-involutive : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (α : X ≃ Y) → ≃-sym (≃-sym α) ＝ α ≃-sym-involutive fe fe' (f , a) = to-subtype-＝ (being-equiv-is-subsingleton fe fe') (inversion-involutive f a) ≃-sym-is-equiv : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → is-equiv (≃-sym {𝓤} {𝓥} {X} {Y}) ≃-sym-is-equiv fe₀ fe₁ fe₂ = invertibles-are-equivs ≃-sym (≃-sym , ≃-sym-involutive fe₀ fe₂ , ≃-sym-involutive fe₁ fe₂) ≃-sym-≃ : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → (X ≃ Y) ≃ (Y ≃ X) ≃-sym-≃ fe₀ fe₁ fe₂ X Y = ≃-sym , ≃-sym-is-equiv fe₀ fe₁ fe₂
Exercise. The hlevels are closed under Σ and, using hfunext, also
under Π. Univalence is not needed, but makes the proof easier. (Without
univalence, we need to show that the hlevels are
closed under equivalence first.)
Π-cong : dfunext 𝓤 𝓥 → dfunext 𝓤 𝓦 → {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {Y' : X → 𝓦 ̇ } → ((x : X) → Y x ≃ Y' x) → Π Y ≃ Π Y' Π-cong fe fe' {X} {Y} {Y'} φ = invertibility-gives-≃ F (G , GF , FG) where f : (x : X) → Y x → Y' x f x = ⌜ φ x ⌝ e : (x : X) → is-equiv (f x) e x = ⌜⌝-is-equiv (φ x) g : (x : X) → Y' x → Y x g x = inverse (f x) (e x) fg : (x : X) (y' : Y' x) → f x (g x y') ＝ y' fg x = inverses-are-sections (f x) (e x) gf : (x : X) (y : Y x) → g x (f x y) ＝ y gf x = inverses-are-retractions (f x) (e x) F : ((x : X) → Y x) → ((x : X) → Y' x) F φ x = f x (φ x) G : ((x : X) → Y' x) → (x : X) → Y x G γ x = g x (γ x) FG : (γ : ((x : X) → Y' x)) → F(G γ) ＝ γ FG γ = fe' (λ x → fg x (γ x)) GF : (φ : ((x : X) → Y x)) → G(F φ) ＝ φ GF φ = fe (λ x → gf x (φ x))
An application of Π-cong is hfunext₂-≃:
hfunext-≃ : hfunext 𝓤 𝓥 → {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (f g : Π A) → (f ＝ g) ≃ (f ∼ g) hfunext-≃ hfe f g = (happly f g , hfe f g) hfunext₂-≃ : hfunext 𝓤 (𝓥 ⊔ 𝓦) → hfunext 𝓥 𝓦 → {X : 𝓤 ̇ } {Y : X → 𝓥 ̇ } {A : (x : X) → Y x → 𝓦 ̇ } (f g : (x : X) (y : Y x) → A x y) → (f ＝ g) ≃ (∀ x y → f x y ＝ g x y) hfunext₂-≃ fe fe' {X} f g = (f ＝ g) ≃⟨ i ⟩ (∀ x → f x ＝ g x) ≃⟨ ii ⟩ (∀ x y → f x y ＝ g x y) ■ where i = hfunext-≃ fe f g ii = Π-cong (hfunext-gives-dfunext fe) (hfunext-gives-dfunext fe) (λ x → hfunext-≃ fe' (f x) (g x)) precomp-invertible : dfunext 𝓥 𝓦 → dfunext 𝓤 𝓦 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) → invertible f → invertible (λ (h : Y → Z) → h ∘ f) precomp-invertible fe fe' {X} {Y} {Z} f (g , η , ε) = (g' , η' , ε') where f' : (Y → Z) → (X → Z) f' h = h ∘ f g' : (X → Z) → (Y → Z) g' k = k ∘ g η' : (h : Y → Z) → g' (f' h) ＝ h η' h = fe (λ y → ap h (ε y)) ε' : (k : X → Z) → f' (g' k) ＝ k ε' k = fe' (λ x → ap k (η x))
We have already proved the following assuming univalence, in order to derive function extensionality from univalence. Now we prove it assuming function extensionality instead.
precomp-is-equiv' : dfunext 𝓥 𝓦 → dfunext 𝓤 𝓦 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) → is-equiv f → is-equiv (λ (h : Y → Z) → h ∘ f) precomp-is-equiv' fe fe' {X} {Y} {Z} f i = invertibles-are-equivs (_∘ f) (precomp-invertible fe fe' f (equivs-are-invertible f i))
More generally, using the half-adjoint condition for equivalences, we have the following dependent version of precomposition, which is also an equivalence assuming function extensionality.
dprecomp : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (A : Y → 𝓦 ̇ ) (f : X → Y) → Π A → Π (A ∘ f) dprecomp A f = _∘ f dprecomp-is-equiv : dfunext 𝓤 𝓦 → dfunext 𝓥 𝓦 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (A : Y → 𝓦 ̇ ) (f : X → Y) → is-equiv f → is-equiv (dprecomp A f) dprecomp-is-equiv fe fe' {X} {Y} A f i = invertibles-are-equivs φ (ψ , ψφ , φψ) where g = inverse f i η = inverses-are-retractions f i ε = inverses-are-sections f i τ : (x : X) → ap f (η x) ＝ ε (f x) τ = half-adjoint-condition f i φ : Π A → Π (A ∘ f) φ = dprecomp A f ψ : Π (A ∘ f) → Π A ψ k y = transport A (ε y) (k (g y)) φψ₀ : (k : Π (A ∘ f)) (x : X) → transport A (ε (f x)) (k (g (f x))) ＝ k x φψ₀ k x = transport A (ε (f x)) (k (g (f x))) ＝⟨ a ⟩ transport A (ap f (η x))(k (g (f x))) ＝⟨ b ⟩ transport (A ∘ f) (η x) (k (g (f x))) ＝⟨ c ⟩ k x ∎ where a = ap (λ - → transport A - (k (g (f x)))) ((τ x)⁻¹) b = (transport-ap A f (η x) (k (g (f x))))⁻¹ c = apd k (η x) φψ : φ ∘ ψ ∼ id φψ k = fe (φψ₀ k) ψφ₀ : (h : Π A) (y : Y) → transport A (ε y) (h (f (g y))) ＝ h y ψφ₀ h y = apd h (ε y) ψφ : ψ ∘ φ ∼ id ψφ h = fe' (ψφ₀ h)
This amounts to saying that we can also change variables in Π:
Π-change-of-variable : dfunext 𝓤 𝓦 → dfunext 𝓥 𝓦 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (A : Y → 𝓦 ̇ ) (f : X → Y) → is-equiv f → (Π y ꞉ Y , A y) ≃ (Π x ꞉ X , A (f x)) Π-change-of-variable fe fe' A f i = dprecomp A f , dprecomp-is-equiv fe fe' A f i
Recall that a function is a Joyal equivalence if it has a section and it has a retraction. We now show that this notion is a subsingleton. For that purpose, we first show that if a function has a retraction then it has at most one section, and that if it has a section then it has at most one retraction.
at-most-one-section : dfunext 𝓥 𝓤 → hfunext 𝓥 𝓥 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → has-retraction f → is-subsingleton (has-section f) at-most-one-section {𝓥} {𝓤} fe hfe {X} {Y} f (g , gf) (h , fh) = d where fe' : dfunext 𝓥 𝓥 fe' = hfunext-gives-dfunext hfe a : invertible f a = joyal-equivs-are-invertible f ((h , fh) , (g , gf)) b : is-singleton (fiber (λ h → f ∘ h) id) b = invertibles-are-equivs (λ h → f ∘ h) (postcomp-invertible fe fe' f a) id r : fiber (λ h → f ∘ h) id → has-section f r (h , p) = (h , happly (f ∘ h) id p) s : has-section f → fiber (λ h → f ∘ h) id s (h , η) = (h , fe' η) rs : (σ : has-section f) → r (s σ) ＝ σ rs (h , η) = to-Σ-＝' q where q : happly (f ∘ h) id (inverse (happly (f ∘ h) id) (hfe (f ∘ h) id) η) ＝ η q = inverses-are-sections (happly (f ∘ h) id) (hfe (f ∘ h) id) η c : is-singleton (has-section f) c = retract-of-singleton (r , s , rs) b d : (σ : has-section f) → h , fh ＝ σ d = singletons-are-subsingletons (has-section f) c (h , fh) at-most-one-retraction : hfunext 𝓤 𝓤 → dfunext 𝓥 𝓤 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → has-section f → is-subsingleton (has-retraction f) at-most-one-retraction {𝓤} {𝓥} hfe fe' {X} {Y} f (g , fg) (h , hf) = d where fe : dfunext 𝓤 𝓤 fe = hfunext-gives-dfunext hfe a : invertible f a = joyal-equivs-are-invertible f ((g , fg) , (h , hf)) b : is-singleton (fiber (λ h → h ∘ f) id) b = invertibles-are-equivs (λ h → h ∘ f) (precomp-invertible fe' fe f a) id r : fiber (λ h → h ∘ f) id → has-retraction f r (h , p) = (h , happly (h ∘ f) id p) s : has-retraction f → fiber (λ h → h ∘ f) id s (h , η) = (h , fe η) rs : (σ : has-retraction f) → r (s σ) ＝ σ rs (h , η) = to-Σ-＝' q where q : happly (h ∘ f) id (inverse (happly (h ∘ f) id) (hfe (h ∘ f) id) η) ＝ η q = inverses-are-sections (happly (h ∘ f) id) (hfe (h ∘ f) id) η c : is-singleton (has-retraction f) c = retract-of-singleton (r , s , rs) b d : (ρ : has-retraction f) → h , hf ＝ ρ d = singletons-are-subsingletons (has-retraction f) c (h , hf) being-joyal-equiv-is-subsingleton : hfunext 𝓤 𝓤 → hfunext 𝓥 𝓥 → dfunext 𝓥 𝓤 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (f : X → Y) → is-subsingleton (is-joyal-equiv f) being-joyal-equiv-is-subsingleton fe₀ fe₁ fe₂ f = ×-is-subsingleton' (at-most-one-section fe₂ fe₁ f , at-most-one-retraction fe₀ fe₂ f)
The fact that a function with a retraction has at most one section can
also be used to prove that the notion of half adjoint equivalence is
property. This is because the type is-hae f is equivalent to the type
Σ (g , ε) ꞉ has-section f , ∀ x → (g (f x) , ε (f x)) ＝ (x , refl (f x)),
where the equality is in the type fiber f (f x).
being-hae-is-subsingleton : dfunext 𝓥 𝓤 → hfunext 𝓥 𝓥 → dfunext 𝓤 (𝓥 ⊔ 𝓤) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-subsingleton (is-hae f) being-hae-is-subsingleton fe₀ fe₁ fe₂ {X} {Y} f = subsingleton-criterion' γ where a = λ g ε x → ((g (f x) , ε (f x)) ＝ (x , refl (f x))) ≃⟨ i g ε x ⟩ (Σ p ꞉ g (f x) ＝ x , transport (λ - → f - ＝ f x) p (ε (f x)) ＝ refl (f x)) ≃⟨ ii g ε x ⟩ (Σ p ꞉ g (f x) ＝ x , ap f p ＝ ε (f x)) ■ where i = λ g ε x → Σ-＝-≃ (g (f x) , ε (f x)) (x , refl (f x)) ii = λ g ε x → Σ-cong (λ p → transport-ap-≃ f p (ε (f x))) b = (Σ (g , ε) ꞉ has-section f , ∀ x → (g (f x) , ε (f x)) ＝ (x , refl (f x))) ≃⟨ i ⟩ (Σ (g , ε) ꞉ has-section f , ∀ x → Σ p ꞉ g (f x) ＝ x , ap f p ＝ ε (f x)) ≃⟨ ii ⟩ (Σ g ꞉ (Y → X) , Σ ε ꞉ f ∘ g ∼ id , ∀ x → Σ p ꞉ g (f x) ＝ x , ap f p ＝ ε (f x)) ≃⟨ iii ⟩ (Σ g ꞉ (Y → X) , Σ ε ꞉ f ∘ g ∼ id , Σ η ꞉ g ∘ f ∼ id , ∀ x → ap f (η x) ＝ ε (f x)) ≃⟨ iv ⟩ is-hae f ■ where i = Σ-cong (λ (g , ε) → Π-cong fe₂ fe₂ (a g ε)) ii = Σ-assoc iii = Σ-cong (λ g → Σ-cong (λ ε → ΠΣ-distr-≃)) iv = Σ-cong (λ g → Σ-flip) γ : is-hae f → is-subsingleton (is-hae f) γ (g₀ , ε₀ , η₀ , τ₀) = i where c : (x : X) → is-set (fiber f (f x)) c x = singletons-are-sets (fiber f (f x)) (haes-are-equivs f (g₀ , ε₀ , η₀ , τ₀) (f x)) d : ((g , ε) : has-section f) → is-subsingleton (∀ x → (g (f x) , ε (f x)) ＝ (x , refl (f x))) d (g , ε) = Π-is-subsingleton fe₂ (λ x → c x (g (f x) , ε (f x)) (x , refl (f x))) e : is-subsingleton (Σ (g , ε) ꞉ has-section f , ∀ x → (g (f x) , ε (f x)) ＝ (x , refl (f x))) e = Σ-is-subsingleton (at-most-one-section fe₀ fe₁ f (g₀ , ε₀)) d i : is-subsingleton (is-hae f) i = equiv-to-subsingleton (≃-sym b) e
Another consequence of function extensionality is that emptiness is a subsingleton:
emptiness-is-subsingleton : dfunext 𝓤 𝓤₀ → (X : 𝓤 ̇ ) → is-subsingleton (is-empty X) emptiness-is-subsingleton fe X f g = fe (λ x → !𝟘 (f x ＝ g x) (f x))
If P is a subsingleton, then so is P + ¬ P. More
generally:
+-is-subsingleton : {P : 𝓤 ̇ } {Q : 𝓥 ̇ } → is-subsingleton P → is-subsingleton Q → (P → Q → 𝟘) → is-subsingleton (P + Q) +-is-subsingleton {𝓤} {𝓥} {P} {Q} i j f = γ where γ : (x y : P + Q) → x ＝ y γ (inl p) (inl p') = ap inl (i p p') γ (inl p) (inr q) = !𝟘 (inl p ＝ inr q) (f p q) γ (inr q) (inl p) = !𝟘 (inr q ＝ inl p) (f p q) γ (inr q) (inr q') = ap inr (j q q') +-is-subsingleton' : dfunext 𝓤 𝓤₀ → {P : 𝓤 ̇ } → is-subsingleton P → is-subsingleton (P + ¬ P) +-is-subsingleton' fe {P} i = +-is-subsingleton i (emptiness-is-subsingleton fe P) (λ p n → n p) EM-is-subsingleton : dfunext (𝓤 ⁺) 𝓤 → dfunext 𝓤 𝓤 → dfunext 𝓤 𝓤₀ → is-subsingleton (EM 𝓤) EM-is-subsingleton fe⁺ fe fe₀ = Π-is-subsingleton fe⁺ (λ P → Π-is-subsingleton fe (λ i → +-is-subsingleton' fe₀ i))
We have been using the mathematical terminology “subsingleton”, but tradition in the formulation of the next notion demands the logical terminology “proposition”. We now discuss the propositional extensionality axiom, its relation to univalence for propositions, and its use for dealing with powersets.
Propositional extensionality says that any two logically equivalent propositions are equal:
propext : ∀ 𝓤 → 𝓤 ⁺ ̇ propext 𝓤 = {P Q : 𝓤 ̇ } → is-prop P → is-prop Q → (P → Q) → (Q → P) → P ＝ Q global-propext : 𝓤ω global-propext = ∀ {𝓤} → propext 𝓤
This is directly implied by univalence:
univalence-gives-propext : is-univalent 𝓤 → propext 𝓤 univalence-gives-propext ua {P} {Q} i j f g = Eq→Id ua P Q γ where γ : P ≃ Q γ = logically-equivalent-subsingletons-are-equivalent P Q i j (f , g)
We also need a version of propositional extensionality for the type
Ω 𝓤 of subsingletons in a given universe 𝓤,
which lives in the next universe:
Ω : (𝓤 : Universe) → 𝓤 ⁺ ̇ Ω 𝓤 = Σ P ꞉ 𝓤 ̇ , is-subsingleton P _holds : Ω 𝓤 → 𝓤 ̇ _holds (P , i) = P holds-is-subsingleton : (p : Ω 𝓤) → is-subsingleton (p holds) holds-is-subsingleton (P , i) = i Ω-ext : dfunext 𝓤 𝓤 → propext 𝓤 → {p q : Ω 𝓤} → (p holds → q holds) → (q holds → p holds) → p ＝ q Ω-ext {𝓤} fe pe {p} {q} f g = to-subtype-＝ (λ _ → being-subsingleton-is-subsingleton fe) (pe (holds-is-subsingleton p) (holds-is-subsingleton q) f g)
With this and Hedberg, we get that Ω is a set:
Ω-is-a-set : dfunext 𝓤 𝓤 → propext 𝓤 → is-set (Ω 𝓤) Ω-is-a-set {𝓤} fe pe = types-with-wconstant-＝-endomaps-are-sets (Ω 𝓤) c where A : (p q : Ω 𝓤) → 𝓤 ̇ A p q = (p holds → q holds) × (q holds → p holds) i : (p q : Ω 𝓤) → is-subsingleton (A p q) i p q = Σ-is-subsingleton (Π-is-subsingleton fe (λ _ → holds-is-subsingleton q)) (λ _ → Π-is-subsingleton fe (λ _ → holds-is-subsingleton p)) g : (p q : Ω 𝓤) → p ＝ q → A p q g p q e = (u , v) where a : p holds ＝ q holds a = ap _holds e u : p holds → q holds u = Id→fun a v : q holds → p holds v = Id→fun (a ⁻¹) h : (p q : Ω 𝓤) → A p q → p ＝ q h p q (u , v) = Ω-ext fe pe u v f : (p q : Ω 𝓤) → p ＝ q → p ＝ q f p q e = h p q (g p q e) k : (p q : Ω 𝓤) (d e : p ＝ q) → f p q d ＝ f p q e k p q d e = ap (h p q) (i p q (g p q d) (g p q e)) c : (p q : Ω 𝓤) → Σ f ꞉ (p ＝ q → p ＝ q), wconstant f c p q = (f p q , k p q)
Hence powersets, even of types that are not sets, are always sets.
powersets-are-sets : hfunext 𝓤 (𝓥 ⁺) → dfunext 𝓥 𝓥 → propext 𝓥 → {X : 𝓤 ̇ } → is-set (X → Ω 𝓥) powersets-are-sets fe fe' pe = Π-is-set fe (λ x → Ω-is-a-set fe' pe)
The above considers X : 𝓤 and Ω 𝓥. When the two universes 𝓤 and
𝓥 are the same, we adopt the usual notation 𝓟 X for the powerset
X → Ω 𝓤 of X.
𝓟 : 𝓤 ̇ → 𝓤 ⁺ ̇ 𝓟 {𝓤} X = X → Ω 𝓤 powersets-are-sets' : Univalence → {X : 𝓤 ̇ } → is-set (𝓟 X) powersets-are-sets' {𝓤} ua = powersets-are-sets (univalence-gives-hfunext' (ua 𝓤) (ua (𝓤 ⁺))) (univalence-gives-dfunext (ua 𝓤)) (univalence-gives-propext (ua 𝓤))
Notice also that both Ω and the powerset live in the next universe. With
propositional resizing, we get
equivalent copies in the same universe.
Membership and containment for elements of the powerset are defined as follows:
_∈_ : {X : 𝓤 ̇ } → X → 𝓟 X → 𝓤 ̇ x ∈ A = A x holds _∉_ : {X : 𝓤 ̇ } → X → 𝓟 X → 𝓤 ̇ x ∉ A = ¬(x ∈ A) _⊆_ : {X : 𝓤 ̇ } → 𝓟 X → 𝓟 X → 𝓤 ̇ A ⊆ B = ∀ x → x ∈ A → x ∈ B ∈-is-subsingleton : {X : 𝓤 ̇ } (A : 𝓟 X) (x : X) → is-subsingleton (x ∈ A) ∈-is-subsingleton A x = holds-is-subsingleton (A x) ⊆-is-subsingleton : dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } (A B : 𝓟 X) → is-subsingleton (A ⊆ B) ⊆-is-subsingleton fe A B = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ _ → ∈-is-subsingleton B x)) ⊆-refl : {X : 𝓤 ̇ } (A : 𝓟 X) → A ⊆ A ⊆-refl A x = 𝑖𝑑 (x ∈ A) ⊆-refl-consequence : {X : 𝓤 ̇ } (A B : 𝓟 X) → A ＝ B → (A ⊆ B) × (B ⊆ A) ⊆-refl-consequence {X} A A (refl A) = ⊆-refl A , ⊆-refl A
Although 𝓟 X is a set even if X is not, the total space
Σ x ꞉ X , A x holds of a member A : 𝓟 X of the powerset need not
be a set. For instance, if A x holds = 𝟙 for all x : X, then the total space is
equivalent to X, which may not be a set.
Propositional and functional extensionality give the usual extensionality condition for the powerset:
subset-extensionality : propext 𝓤 → dfunext 𝓤 𝓤 → dfunext 𝓤 (𝓤 ⁺) → {X : 𝓤 ̇ } {A B : 𝓟 X} → A ⊆ B → B ⊆ A → A ＝ B subset-extensionality pe fe fe' {X} {A} {B} h k = fe' φ where φ : (x : X) → A x ＝ B x φ x = to-subtype-＝ (λ _ → being-subsingleton-is-subsingleton fe) (pe (holds-is-subsingleton (A x)) (holds-is-subsingleton (B x)) (h x) (k x))
And hence so does univalence:
subset-extensionality' : Univalence → {X : 𝓤 ̇ } {A B : 𝓟 X} → A ⊆ B → B ⊆ A → A ＝ B subset-extensionality' {𝓤} ua = subset-extensionality (univalence-gives-propext (ua 𝓤)) (univalence-gives-dfunext (ua 𝓤)) (univalence-gives-dfunext' (ua 𝓤) (ua (𝓤 ⁺)))
For set-level mathematics, function extensionality and propositional extensionality are often the only consequences of univalence that are needed. A noteworthy exception is the theorem that the type of ordinals in a universe is an ordinal in the next universe, which requires univalence for sets (see the HoTT book or this).
This section is not needed anywhere else and can be safely skipped. We characterize propositional univalence in terms of propositional extensionality and a restricted form of function extensionality.
By propositional univalence mean any of the following two equivalent notions.
prop-univalence prop-univalence' : (𝓤 : Universe) → 𝓤 ⁺ ̇ prop-univalence 𝓤 = (A : 𝓤 ̇ ) → is-prop A → (X : 𝓤 ̇ ) → is-equiv (Id→Eq A X) prop-univalence' 𝓤 = (A : 𝓤 ̇ ) → is-prop A → (X : 𝓤 ̇ ) → is-prop X → is-equiv (Id→Eq A X) prop-univalence-agreement : prop-univalence' 𝓤 ⇔ prop-univalence 𝓤 prop-univalence-agreement = (λ pu' A i X e → pu' A i X (equiv-to-subsingleton (≃-sym e) i) e) , (λ pu A i X _ → pu A i X)
The restricted form of function extensionality is given by any of the following four notions, which are equivalent under propositional extensionality and have the following type:
props-form-exponential-ideal props-are-closed-under-Π prop-vvfunext prop-hfunext : ∀ 𝓤 → 𝓤 ⁺ ̇
They are defined as follows.
props-form-exponential-ideal 𝓤 = (X A : 𝓤 ̇ ) → is-prop A → is-prop (X → A) props-are-closed-under-Π 𝓤 = {X : 𝓤 ̇ } {A : X → 𝓤 ̇ } → is-prop X → ((x : X) → is-prop (A x)) → is-prop (Π A) prop-vvfunext 𝓤 = {X : 𝓤 ̇ } {A : X → 𝓤 ̇ } → is-prop X → ((x : X) → is-singleton (A x)) → is-singleton (Π A) prop-hfunext 𝓤 = {X : 𝓤 ̇ } {A : X → 𝓤 ̇ } → is-prop X → (f g : Π A) → is-equiv (happly f g)
The last three agree unconditionally:
first-propositional-function-extensionality-agreement : (props-are-closed-under-Π 𝓤 → prop-vvfunext 𝓤) × (prop-vvfunext 𝓤 → prop-hfunext 𝓤) × (prop-hfunext 𝓤 → props-are-closed-under-Π 𝓤)
The proof is postponed. Moreover, as already mentioned, the four of them agree under propositional extensionality.
second-propositional-function-extensionality-agreement : propext 𝓤 → (props-form-exponential-ideal 𝓤 ⇔ props-are-closed-under-Π 𝓤)
The proof is again postponed, and this time there is a twist: we use a detour via propositional univalence to prove this.
Question. Does propositional function extensionality, in any of the above four incarnations, perhaps in the presence of propositional extensionality, imply full function extensionality?
Discussion. Notice that props-form-exponential-ideal requires A to be a proposition, but not X, whereas prop-hfunext requires X to be a proposition but not A x, while the other two versions require both X and A(x) to be propositions, and yet all versions are equivalent under propositional extensionality. Given this, a positive answer to the above question is not unlikely. We leave this as an open problem.
The following is the main theorem of this section, where the notion props-form-an-exponential-ideal plays a crucial role in its proof.
characterization-of-propositional-univalence : prop-univalence 𝓤 ⇔ (propext 𝓤 × props-are-closed-under-Π 𝓤)
The remainder of this section is devoted to prove this, which has the second propositional function extensionality agreement as a consequence, but relies on the first agreement.
It is direct that propositional univalence implies propositional extensionality.
prop-univalence-gives-propext : prop-univalence 𝓤 → propext 𝓤 prop-univalence-gives-propext pu {P} {Q} i j f g = δ where γ : P ≃ Q γ = logically-equivalent-subsingletons-are-equivalent P Q i j (f , g) δ : P ＝ Q δ = inverse (Id→Eq P Q) (pu P i Q) γ
To show that propositional univalence implies that the propositions form an exponential ideal, we first need some lemmas.
prop-≃-induction : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥)⁺ ̇ prop-≃-induction 𝓤 𝓥 = (P : 𝓤 ̇ ) → is-prop P → (A : (X : 𝓤 ̇ ) → P ≃ X → 𝓥 ̇ ) → A P (id-≃ P) → (X : 𝓤 ̇ ) (e : P ≃ X) → A X e prop-J-equiv : prop-univalence 𝓤 → (𝓥 : Universe) → prop-≃-induction 𝓤 𝓥 prop-J-equiv {𝓤} pu 𝓥 P i A a X e = γ where A' : (X : 𝓤 ̇ ) → P ＝ X → 𝓥 ̇ A' X q = A X (Id→Eq P X q) f : (X : 𝓤 ̇ ) (q : P ＝ X) → A' X q f = ℍ P A' a r : P ＝ X r = inverse (Id→Eq P X) (pu P i X) e g : A X (Id→Eq P X r) g = f X r γ : A X (id e) γ = transport (A X) (inverses-are-sections (Id→Eq P X) (pu P i X) e) g prop-precomp-is-equiv : prop-univalence 𝓤 → (X Y Z : 𝓤 ̇ ) → is-prop X → (f : X → Y) → is-equiv f → is-equiv (λ (g : Y → Z) → g ∘ f) prop-precomp-is-equiv {𝓤} pu X Y Z i f f-is-equiv = prop-J-equiv pu 𝓤 X i (λ _ e → is-equiv (λ g → g ∘ ⌜ e ⌝)) (id-is-equiv (X → Z)) Y (f , f-is-equiv)
We now apply the above lemmas to adapt the above proof that univalence implies function extensionality in order to obtain the following.
prop-univalence-gives-props-form-exponential-ideal : prop-univalence 𝓤 → props-form-exponential-ideal 𝓤 prop-univalence-gives-props-form-exponential-ideal {𝓤} pu X A A-is-prop = γ where Δ : 𝓤 ̇ Δ = Σ a₀ ꞉ A , Σ a₁ ꞉ A , a₀ ＝ a₁ δ : A → Δ δ a = (a , a , refl a) π₀ π₁ : Δ → A π₀ (a₀ , a₁ , a) = a₀ π₁ (a₀ , a₁ , a) = a₁ δ-is-equiv : is-equiv δ δ-is-equiv = invertibles-are-equivs δ (π₀ , η , ε) where η : (a : A) → π₀ (δ a) ＝ a η a = refl a ε : (d : Δ) → δ (π₀ d) ＝ d ε (a , a , refl a) = refl (a , a , refl a) φ : (Δ → A) → (A → A) φ π = π ∘ δ φ-is-equiv : is-equiv φ φ-is-equiv = prop-precomp-is-equiv pu A Δ A A-is-prop δ δ-is-equiv p : φ π₀ ＝ φ π₁ p = refl (𝑖𝑑 A) q : π₀ ＝ π₁ q = equivs-are-lc φ φ-is-equiv p h : (f₀ f₁ : X → A) → f₀ ∼ f₁ h f₀ f₁ x = A-is-prop (f₀ x) (f₁ x) γ : (f₀ f₁ : X → A) → f₀ ＝ f₁ γ f₀ f₁ = ap (λ π x → π (f₀ x , f₁ x , h f₀ f₁ x)) q
To prove the converse, we first establish the first propositional function extensionality agreement, in a number of steps. For this purpose we adapt some proofs regarding the various notions of (full) function extensionality.
props-are-closed-under-Π-gives-prop-vvfunext : props-are-closed-under-Π 𝓤 → prop-vvfunext 𝓤 props-are-closed-under-Π-gives-prop-vvfunext c {X} {A} X-is-prop A-is-prop-valued = γ where f : Π A f x = center (A x) (A-is-prop-valued x) d : (g : Π A) → f ＝ g d = c X-is-prop (λ (x : X) → singletons-are-subsingletons (A x) (A-is-prop-valued x)) f γ : is-singleton (Π A) γ = f , d prop-vvfunext-gives-prop-hfunext : prop-vvfunext 𝓤 → prop-hfunext 𝓤 prop-vvfunext-gives-prop-hfunext vfe {X} {Y} X-is-prop f = γ where a : (x : X) → is-singleton (Σ y ꞉ Y x , f x ＝ y) a x = singleton-types'-are-singletons (Y x) (f x) c : is-singleton (Π x ꞉ X , Σ y ꞉ Y x , f x ＝ y) c = vfe X-is-prop a ρ : (Σ g ꞉ Π Y , f ∼ g) ◁ (Π x ꞉ X , Σ y ꞉ Y x , f x ＝ y) ρ = ≃-gives-▷ ΠΣ-distr-≃ d : is-singleton (Σ g ꞉ Π Y , f ∼ g) d = retract-of-singleton ρ c e : (Σ g ꞉ Π Y , f ＝ g) → (Σ g ꞉ Π Y , f ∼ g) e = NatΣ (happly f) i : is-equiv e i = maps-of-singletons-are-equivs e (singleton-types'-are-singletons (Π Y) f) d γ : (g : Π Y) → is-equiv (happly f g) γ = NatΣ-equiv-gives-fiberwise-equiv (happly f) i prop-hfunext-gives-props-are-closed-under-Π : prop-hfunext 𝓤 → props-are-closed-under-Π 𝓤 prop-hfunext-gives-props-are-closed-under-Π hfe {X} {A} X-is-prop A-is-prop-valued f g = γ where γ : f ＝ g γ = inverse (happly f g) (hfe X-is-prop f g) (λ x → A-is-prop-valued x (f x) (g x))
With this we can now fulfill the first promise:
first-propositional-function-extensionality-agreement = props-are-closed-under-Π-gives-prop-vvfunext , prop-vvfunext-gives-prop-hfunext , prop-hfunext-gives-props-are-closed-under-Π
We have already proved the following lemma being-prop-is-prop, as being-subsingleton-is-subsingleton, but under the stronger assumption vvfunext. The proof is almost, but not literally, the same, and needs some of the above lemmas.
prop-vvfunext-gives-props-are-closed-under-Π : prop-vvfunext 𝓤 → props-are-closed-under-Π 𝓤 prop-vvfunext-gives-props-are-closed-under-Π vfe = prop-hfunext-gives-props-are-closed-under-Π (prop-vvfunext-gives-prop-hfunext vfe) being-prop-is-prop : prop-vvfunext 𝓤 → {X : 𝓤 ̇ } → is-prop (is-prop X) being-prop-is-prop vfe {X} i j = γ where k : is-set X k = subsingletons-are-sets X i a : (x y : X) → i x y ＝ j x y a x y = k x y (i x y) (j x y) b : (x : X) → i x ＝ j x b x = prop-vvfunext-gives-props-are-closed-under-Π vfe i (subsingletons-are-sets X i x) (i x) (j x) c : (x : X) → is-prop ((y : X) → x ＝ y) c x = singletons-are-subsingletons ((y : X) → x ＝ y) (vfe i (λ y → pointed-subsingletons-are-singletons (x ＝ y) (i x y) (k x y))) γ : i ＝ j γ = prop-vvfunext-gives-props-are-closed-under-Π vfe i c i j
Similarly, we have already proved the following, as being-singleton-is-subsingleton, under the stronger assumption dfunext.
being-singleton-is-prop : props-are-closed-under-Π 𝓤 → {X : 𝓤 ̇ } → is-prop (is-singleton X) being-singleton-is-prop c {X} (x , φ) (y , γ) = p where i : is-subsingleton X i = singletons-are-subsingletons X (y , γ) s : is-set X s = subsingletons-are-sets X i a : (z : X) → is-subsingleton ((t : X) → z ＝ t) a z = c i (λ x → s z x) b : x ＝ y b = φ y p : (x , φ) ＝ (y , γ) p = to-subtype-＝ a b
The following hasn’t occurred above in any form, and is crucial for the main theorem of this section.
Id-of-props-is-prop : propext 𝓤 → prop-vvfunext 𝓤 → (P : 𝓤 ̇ ) → is-prop P → (X : 𝓤 ̇ ) → is-prop (P ＝ X) Id-of-props-is-prop {𝓤} pe vfe P i = Hedberg P (λ X → h X , k X) where module _ (X : 𝓤 ̇ ) where f : P ＝ X → is-prop X × (P ⇔ X) f p = transport is-prop p i , Id→fun p , (Id→fun (p ⁻¹)) g : is-prop X × (P ⇔ X) → P ＝ X g (l , φ , ψ) = pe i l φ ψ h : P ＝ X → P ＝ X h = g ∘ f j : is-prop (is-prop X × (P ⇔ X)) j = ×-is-subsingleton' ((λ (_ : P ⇔ X) → being-prop-is-prop vfe) , (λ (l : is-prop X) → ×-is-subsingleton (prop-vvfunext-gives-props-are-closed-under-Π vfe i (λ _ → l)) (prop-vvfunext-gives-props-are-closed-under-Π vfe l (λ _ → i)))) k : wconstant h k p q = ap g (j (f p) (f q)) being-equiv-of-props-is-prop : props-are-closed-under-Π 𝓤 → {X Y : 𝓤 ̇ } → is-prop X → is-prop Y → (f : X → Y) → is-prop (is-equiv f) being-equiv-of-props-is-prop c i j f = c j (λ y → being-singleton-is-prop c)
Armed with the above lemmas, we are now in a position to show that propositional extensionality and propositional function extensionality together imply propositional univalence.
propext-and-props-are-closed-under-Π-give-prop-univalence : propext 𝓤 → props-are-closed-under-Π 𝓤 → prop-univalence 𝓤 propext-and-props-are-closed-under-Π-give-prop-univalence pe c A i X = γ where l : A ≃ X → is-subsingleton X l e = equiv-to-subsingleton (≃-sym e) i eqtoid : A ≃ X → A ＝ X eqtoid e = pe i (equiv-to-subsingleton (≃-sym e) i) ⌜ e ⌝ ⌜ ≃-sym e ⌝ m : is-subsingleton (A ≃ X) m (f₀ , k₀) (f₁ , k₁) = δ where j : (f : A → X) → is-prop (is-equiv f) j = being-equiv-of-props-is-prop c i (equiv-to-subsingleton (≃-sym (f₀ , k₀)) i) p : f₀ ＝ f₁ p = c i (λ (a : A) → l (f₁ , k₁)) f₀ f₁ δ : (f₀ , k₀) ＝ (f₁ , k₁) δ = to-subtype-＝ j p ε : (e : A ≃ X) → Id→Eq A X (eqtoid e) ＝ e ε e = m (Id→Eq A X (eqtoid e)) e η : (q : A ＝ X) → eqtoid (Id→Eq A X q) ＝ q η q = Id-of-props-is-prop pe (props-are-closed-under-Π-gives-prop-vvfunext c) A i X (eqtoid (Id→Eq A X q)) q γ : is-equiv (Id→Eq A X) γ = invertibles-are-equivs (Id→Eq A X) (eqtoid , η , ε)
We now need some lemmas to prove the converse. The next two have already been proved under different hypotheses.
prop-postcomp-invertible : {X Y A : 𝓤 ̇ } → props-form-exponential-ideal 𝓤 → is-prop X → is-prop Y → (f : X → Y) → invertible f → invertible (λ (h : A → X) → f ∘ h) prop-postcomp-invertible {𝓤} {X} {Y} {A} pei i j f (g , η , ε) = γ where f' : (A → X) → (A → Y) f' h = f ∘ h g' : (A → Y) → (A → X) g' k = g ∘ k η' : (h : A → X) → g' (f' h) ＝ h η' h = pei A X i (g' (f' h)) h ε' : (k : A → Y) → f' (g' k) ＝ k ε' k = pei A Y j (f' (g' k)) k γ : invertible f' γ = (g' , η' , ε') prop-postcomp-is-equiv : {X Y A : 𝓤 ̇ } → props-form-exponential-ideal 𝓤 → is-prop X → is-prop Y → (f : X → Y) → is-equiv f → is-equiv (λ (h : A → X) → f ∘ h) prop-postcomp-is-equiv pei i j f e = invertibles-are-equivs (λ h → f ∘ h) (prop-postcomp-invertible pei i j f (equivs-are-invertible f e))
The following shows that props-form-exponential-ideal is stronger than prop-vvfunext (and perhaps properly stronger in the absence of propositional extensionality).
props-form-exponential-ideal-gives-vvfunext : props-form-exponential-ideal 𝓤 → prop-vvfunext 𝓤 props-form-exponential-ideal-gives-vvfunext {𝓤} pei {X} {A} X-is-prop φ = γ where f : Σ A → X f = pr₁ A-is-prop-valued : (x : X) → is-prop (A x) A-is-prop-valued x = singletons-are-subsingletons (A x) (φ x) k : is-prop (Σ A) k = Σ-is-subsingleton X-is-prop A-is-prop-valued f-is-equiv : is-equiv f f-is-equiv = pr₁-is-equiv φ g : (X → Σ A) → (X → X) g h = f ∘ h e : is-equiv g e = prop-postcomp-is-equiv pei k X-is-prop f f-is-equiv i : is-singleton (Σ h ꞉ (X → Σ A), f ∘ h ＝ 𝑖𝑑 X) i = e (𝑖𝑑 X) r : (Σ h ꞉ (X → Σ A), f ∘ h ＝ 𝑖𝑑 X) → Π A r (h , p) x = transport A (happly (f ∘ h) (𝑖𝑑 X) p x) (pr₂ (h x)) s : Π A → (Σ h ꞉ (X → Σ A), f ∘ h ＝ 𝑖𝑑 X) s ψ = (λ x → x , ψ x) , refl (𝑖𝑑 X) η : ∀ ψ → r (s ψ) ＝ ψ η ψ = refl (r (s ψ)) γ : is-singleton (Π A) γ = retract-of-singleton (r , s , η) i
Here are some corollaries of the above.
prop-univalence-gives-props-are-closed-under-Π : prop-univalence 𝓤 → props-are-closed-under-Π 𝓤 prop-univalence-gives-props-are-closed-under-Π pu = prop-vvfunext-gives-props-are-closed-under-Π (props-form-exponential-ideal-gives-vvfunext (prop-univalence-gives-props-form-exponential-ideal pu)) propext-and-props-closed-under-Π-give-props-form-exponential-ideal : propext 𝓤 → props-are-closed-under-Π 𝓤 → props-form-exponential-ideal 𝓤 propext-and-props-closed-under-Π-give-props-form-exponential-ideal pe c = prop-univalence-gives-props-form-exponential-ideal (propext-and-props-are-closed-under-Π-give-prop-univalence pe c) props-form-exponential-ideal-gives-props-are-closed-under-Π : props-form-exponential-ideal 𝓤 → props-are-closed-under-Π 𝓤 props-form-exponential-ideal-gives-props-are-closed-under-Π pei = prop-vvfunext-gives-props-are-closed-under-Π (props-form-exponential-ideal-gives-vvfunext pei)
And with this we can complete the proof of the main theorem of this section, formulated above in Agda.
characterization-of-propositional-univalence {𝓤} = α , β where α : prop-univalence 𝓤 → propext 𝓤 × props-are-closed-under-Π 𝓤 α pu = prop-univalence-gives-propext pu , prop-univalence-gives-props-are-closed-under-Π pu β : propext 𝓤 × props-are-closed-under-Π 𝓤 → prop-univalence 𝓤 β (pe , fe) = propext-and-props-are-closed-under-Π-give-prop-univalence pe fe
And we can also fulfill the second promise again using prop-univalence-gives-props-are-exponential-ideal.
second-propositional-function-extensionality-agreement {𝓤} pe = props-form-exponential-ideal-gives-props-are-closed-under-Π , propext-and-props-closed-under-Π-give-props-form-exponential-ideal pe
We first prove some properties of equivalence symmetrization and composition:
id-≃-left : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (α : X ≃ Y) → id-≃ X ● α ＝ α id-≃-left fe fe' α = to-subtype-＝ (being-equiv-is-subsingleton fe fe') (refl _) ≃-sym-left-inverse : dfunext 𝓥 𝓥 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (α : X ≃ Y) → ≃-sym α ● α ＝ id-≃ Y ≃-sym-left-inverse fe (f , e) = to-subtype-＝ (being-equiv-is-subsingleton fe fe) p where p : f ∘ inverse f e ＝ id p = fe (inverses-are-sections f e) ≃-sym-right-inverse : dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (α : X ≃ Y) → α ● ≃-sym α ＝ id-≃ X ≃-sym-right-inverse fe (f , e) = to-subtype-＝ (being-equiv-is-subsingleton fe fe) p where p : inverse f e ∘ f ＝ id p = fe (inverses-are-retractions f e)
We then transfer the above to equivalence types:
≃-Sym : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X ≃ Y) ≃ (Y ≃ X) ≃-Sym fe₀ fe₁ fe₂ = ≃-sym , ≃-sym-is-equiv fe₀ fe₁ fe₂ ≃-Comp : dfunext 𝓦 (𝓥 ⊔ 𝓦) → dfunext (𝓥 ⊔ 𝓦) (𝓥 ⊔ 𝓦 ) → dfunext 𝓥 𝓥 → dfunext 𝓦 (𝓤 ⊔ 𝓦) → dfunext (𝓤 ⊔ 𝓦) (𝓤 ⊔ 𝓦 ) → dfunext 𝓤 𝓤 → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (Z : 𝓦 ̇ ) → X ≃ Y → (Y ≃ Z) ≃ (X ≃ Z) ≃-Comp fe₀ fe₁ fe₂ fe₃ fe₄ fe₅ Z α = invertibility-gives-≃ (α ●_) ((≃-sym α ●_) , p , q) where p = λ β → ≃-sym α ● (α ● β) ＝⟨ ●-assoc fe₀ fe₁ (≃-sym α) α β ⟩ (≃-sym α ● α) ● β ＝⟨ ap (_● β) (≃-sym-left-inverse fe₂ α) ⟩ id-≃ _ ● β ＝⟨ id-≃-left fe₀ fe₁ _ ⟩ β ∎ q = λ γ → α ● (≃-sym α ● γ) ＝⟨ ●-assoc fe₃ fe₄ α (≃-sym α) γ ⟩ (α ● ≃-sym α) ● γ ＝⟨ ap (_● γ) (≃-sym-right-inverse fe₅ α) ⟩ id-≃ _ ● γ ＝⟨ id-≃-left fe₃ fe₄ _ ⟩ γ ∎
Using this we get the following self-congruence property of equivalences:
Eq-Eq-cong' : dfunext 𝓥 (𝓤 ⊔ 𝓥) → dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) → dfunext 𝓤 𝓤 → dfunext 𝓥 (𝓥 ⊔ 𝓦) → dfunext (𝓥 ⊔ 𝓦) (𝓥 ⊔ 𝓦) → dfunext 𝓦 𝓦 → dfunext 𝓦 (𝓥 ⊔ 𝓦) → dfunext 𝓥 𝓥 → dfunext 𝓦 (𝓦 ⊔ 𝓣) → dfunext (𝓦 ⊔ 𝓣) (𝓦 ⊔ 𝓣) → dfunext 𝓣 𝓣 → dfunext 𝓣 (𝓦 ⊔ 𝓣) → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : 𝓦 ̇ } {B : 𝓣 ̇ } → X ≃ A → Y ≃ B → (X ≃ Y) ≃ (A ≃ B) Eq-Eq-cong' fe₀ fe₁ fe₂ fe₃ fe₄ fe₅ fe₆ fe₇ fe₈ fe₉ fe₁₀ fe₁₁ {X} {Y} {A} {B} α β = X ≃ Y ≃⟨ ≃-Comp fe₀ fe₁ fe₂ fe₃ fe₄ fe₅ Y (≃-sym α) ⟩ A ≃ Y ≃⟨ ≃-Sym fe₃ fe₆ fe₄ ⟩ Y ≃ A ≃⟨ ≃-Comp fe₆ fe₄ fe₇ fe₈ fe₉ fe₁₀ A (≃-sym β) ⟩ B ≃ A ≃⟨ ≃-Sym fe₈ fe₁₁ fe₉ ⟩ A ≃ B ■
The above shows why global function extensionality would be a better assumption in practice, and so we define:
Eq-Eq-cong : global-dfunext → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {A : 𝓦 ̇ } {B : 𝓣 ̇ } → X ≃ A → Y ≃ B → (X ≃ Y) ≃ (A ≃ B) Eq-Eq-cong fe = Eq-Eq-cong' fe fe fe fe fe fe fe fe fe fe fe fe
A function is called an embedding if its fibers are all subsingletons. In particular, equivalences are embeddings. However, sections of types more general than sets don’t need to be embeddings.
is-embedding : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ is-embedding f = (y : codomain f) → is-subsingleton (fiber f y)
This says that for every y : Y there is at most one x : X with f
x ＝ y, or, more precisely, there is at most one pair (x , p)
with x : X and p : f x ＝ y.
being-embedding-is-subsingleton : global-dfunext → {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-subsingleton (is-embedding f) being-embedding-is-subsingleton fe f = Π-is-subsingleton fe (λ x → being-subsingleton-is-subsingleton fe)
For example, if A is a subsingleton, then the second projection A ×
X → X is an embedding:
pr₂-embedding : (A : 𝓤 ̇ ) (X : 𝓥 ̇ ) → is-subsingleton A → is-embedding (λ (z : A × X) → pr₂ z) pr₂-embedding A X i x ((a , x) , refl x) ((b , x) , refl x) = p where p : ((a , x) , refl x) ＝ ((b , x) , refl x) p = ap (λ - → ((- , x) , refl x)) (i a b)
Exercise. Show that the converse of pr₂-embedding holds.
More generally, with the arguments swapped, the projection Σ A → X
is an embedding if A x is a subsingleton for every x : X:
pr₁-is-embedding : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ((x : X) → is-subsingleton (A x)) → is-embedding (λ (σ : Σ A) → pr₁ σ) pr₁-is-embedding i x ((x , a) , refl x) ((x , a') , refl x) = γ where p : a ＝ a' p = i x a a' γ : (x , a) , refl x ＝ (x , a') , refl x γ = ap (λ - → (x , -) , refl x) p
Exercise. Show that the converse of pr₁-is-embedding holds.
equivs-are-embeddings : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f → is-embedding f equivs-are-embeddings f i y = singletons-are-subsingletons (fiber f y) (i y) id-is-embedding : {X : 𝓤 ̇ } → is-embedding (𝑖𝑑 X) id-is-embedding {𝓤} {X} = equivs-are-embeddings id (id-is-equiv X) ∘-embedding : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } {f : X → Y} {g : Y → Z} → is-embedding g → is-embedding f → is-embedding (g ∘ f) ∘-embedding {𝓤} {𝓥} {𝓦} {X} {Y} {Z} {f} {g} d e = h where A : (z : Z) → 𝓤 ⊔ 𝓥 ⊔ 𝓦 ̇ A z = Σ (y , p) ꞉ fiber g z , fiber f y i : (z : Z) → is-subsingleton (A z) i z = Σ-is-subsingleton (d z) (λ w → e (pr₁ w)) φ : (z : Z) → fiber (g ∘ f) z → A z φ z (x , p) = (f x , p) , x , refl (f x) γ : (z : Z) → A z → fiber (g ∘ f) z γ z ((_ , p) , x , refl _) = x , p η : (z : Z) (t : fiber (g ∘ f) z) → γ z (φ z t) ＝ t η _ (x , refl _) = refl (x , refl ((g ∘ f) x)) h : (z : Z) → is-subsingleton (fiber (g ∘ f) z) h z = lc-maps-reflect-subsingletons (φ z) (sections-are-lc (φ z) (γ z , η z)) (i z)
We can use the following criterion to prove that some maps are embeddings:
embedding-lemma : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → ((x : X) → is-singleton (fiber f (f x))) → is-embedding f embedding-lemma f φ = γ where γ : (y : codomain f) (u v : fiber f y) → u ＝ v γ y (x , p) v = j (x , p) v where q : fiber f (f x) ＝ fiber f y q = ap (fiber f) p i : is-singleton (fiber f y) i = transport is-singleton q (φ x) j : is-subsingleton (fiber f y) j = singletons-are-subsingletons (fiber f y) i embedding-criterion : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → ((x x' : X) → (f x ＝ f x') ≃ (x ＝ x')) → is-embedding f embedding-criterion f e = embedding-lemma f b where X = domain f a : (x : X) → (Σ x' ꞉ X , f x' ＝ f x) ≃ (Σ x' ꞉ X , x' ＝ x) a x = Σ-cong (λ x' → e x' x) a' : (x : X) → fiber f (f x) ≃ singleton-type x a' = a b : (x : X) → is-singleton (fiber f (f x)) b x = equiv-to-singleton (a' x) (singleton-types-are-singletons X x)
An equivalent formulation of f being an embedding is that the map
ap f {x} {x'} : x ＝ x' → f x ＝ f x'
is an equivalence for all x x' : X.
ap-is-equiv-gives-embedding : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → ((x x' : X) → is-equiv (ap f {x} {x'})) → is-embedding f ap-is-equiv-gives-embedding f i = embedding-criterion f (λ x' x → ≃-sym (ap f {x'} {x} , i x' x)) embedding-gives-ap-is-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-embedding f → (x x' : X) → is-equiv (ap f {x} {x'}) embedding-gives-ap-is-equiv {𝓤} {𝓥} {X} f e = γ where d : (x' : X) → (Σ x ꞉ X , f x' ＝ f x) ≃ (Σ x ꞉ X , f x ＝ f x') d x' = Σ-cong (λ x → ⁻¹-≃ (f x') (f x)) s : (x' : X) → is-subsingleton (Σ x ꞉ X , f x' ＝ f x) s x' = equiv-to-subsingleton (d x') (e (f x')) γ : (x x' : X) → is-equiv (ap f {x} {x'}) γ x = singleton-equiv-lemma x (λ x' → ap f {x} {x'}) (pointed-subsingletons-are-singletons (Σ x' ꞉ X , f x ＝ f x') (x , (refl (f x))) (s x)) embedding-criterion-converse : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-embedding f → ((x' x : X) → (f x' ＝ f x) ≃ (x' ＝ x)) embedding-criterion-converse f e x' x = ≃-sym (ap f {x'} {x} , embedding-gives-ap-is-equiv f e x' x)
Hence embeddings of arbitrary types are left cancellable, but the converse fails in general.
embeddings-are-lc : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-embedding f → left-cancellable f embeddings-are-lc f e {x} {y} = ⌜ embedding-criterion-converse f e x y ⌝
Conversely, left cancellable maps into sets are always embeddings.
lc-maps-into-sets-are-embeddings : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → left-cancellable f → is-set Y → is-embedding f lc-maps-into-sets-are-embeddings {𝓤} {𝓥} {X} {Y} f lc i y = γ where γ : is-subsingleton (Σ x ꞉ X , f x ＝ y) γ (x , p) (x' , p') = to-subtype-＝ j q where j : (x : X) → is-subsingleton (f x ＝ y) j x = i (f x) y q : x ＝ x' q = lc (f x ＝⟨ p ⟩ y ＝⟨ p' ⁻¹ ⟩ f x' ∎)
If an embedding has a section, then it is an equivalence.
embedding-with-section-is-equiv : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-embedding f → has-section f → is-equiv f embedding-with-section-is-equiv f i (g , η) y = pointed-subsingletons-are-singletons (fiber f y) (g y , η y) (i y)
Later we will see that a necessary and sufficient condition for an embedding to be an equivalence is that it is a surjection.
If a type Y is embedded into Z, then the function type X → Y is
embedded into X → Z. More generally, if A x is embedded into B x
for every x : X, then the dependent function type Π A is embedded
into Π B.
NatΠ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → Nat A B → Π A → Π B NatΠ τ f x = τ x (f x)
(Notice that NatΠ is a dependently typed version of the combinator S from
combinatory logic. Its logical interpretation, here, is that if A x implies B x for all x : X, and A x holds for all x : X, then B x holds for all x : X too.)
NatΠ-is-embedding : hfunext 𝓤 𝓥 → hfunext 𝓤 𝓦 → {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → (τ : Nat A B) → ((x : X) → is-embedding (τ x)) → is-embedding (NatΠ τ) NatΠ-is-embedding v w {X} {A} τ i = embedding-criterion (NatΠ τ) γ where γ : (f g : Π A) → (NatΠ τ f ＝ NatΠ τ g) ≃ (f ＝ g) γ f g = (NatΠ τ f ＝ NatΠ τ g) ≃⟨ hfunext-≃ w (NatΠ τ f) (NatΠ τ g) ⟩ (NatΠ τ f ∼ NatΠ τ g) ≃⟨ b ⟩ (f ∼ g) ≃⟨ ≃-sym (hfunext-≃ v f g) ⟩ (f ＝ g) ■ where a : (x : X) → (NatΠ τ f x ＝ NatΠ τ g x) ≃ (f x ＝ g x) a x = embedding-criterion-converse (τ x) (i x) (f x) (g x) b : (NatΠ τ f ∼ NatΠ τ g) ≃ (f ∼ g) b = Π-cong (hfunext-gives-dfunext w) (hfunext-gives-dfunext v) a
Postcomposition with an embedding is itself an embedding (of a
function type into another). This amounts to saying that any function
f : X → A and any embedding g : Y → A can be completed to a
commutative triangle in at most one way:
triangle-lemma : dfunext 𝓦 (𝓤 ⊔ 𝓥) → {Y : 𝓤 ̇ } {A : 𝓥 ̇ } (g : Y → A) → is-embedding g → {X : 𝓦 ̇ } (f : X → A) → is-subsingleton (Σ h ꞉ (X → Y) , g ∘ h ∼ f) triangle-lemma fe {Y} {A} g i {X} f = iv where ii : (x : X) → is-subsingleton (Σ y ꞉ Y , g y ＝ f x) ii x = i (f x) iii : is-subsingleton (Π x ꞉ X , Σ y ꞉ Y , g y ＝ f x) iii = Π-is-subsingleton fe ii iv : is-subsingleton (Σ h ꞉ (X → Y) , g ∘ h ∼ f) iv = equiv-to-subsingleton (≃-sym ΠΣ-distr-≃) iii postcomp-is-embedding : dfunext 𝓦 (𝓤 ⊔ 𝓥) → hfunext 𝓦 𝓥 → {Y : 𝓤 ̇ } {A : 𝓥 ̇ } (g : Y → A) → is-embedding g → (X : 𝓦 ̇ ) → is-embedding (λ (h : X → Y) → g ∘ h) postcomp-is-embedding fe hfe {Y} {A} g i X = γ where γ : (f : X → A) → is-subsingleton (Σ h ꞉ (X → Y) , g ∘ h ＝ f) γ f = equiv-to-subsingleton u (triangle-lemma fe g i f) where u : (Σ h ꞉ (X → Y) , g ∘ h ＝ f) ≃ (Σ h ꞉ (X → Y) , g ∘ h ∼ f) u = Σ-cong (λ h → hfunext-≃ hfe (g ∘ h) f)
We conclude this section by introducing notation for the type of embeddings.
_↪_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ X ↪ Y = Σ f ꞉ (X → Y), is-embedding f Emb→fun : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X ↪ Y) → (X → Y) Emb→fun (f , i) = f
The following justifies the terminology subsingleton:
Exercise. (1)
Show that the type is-subsingleton X is logically equivalent to the type X ↪ 𝟙. (2)
Hence assuming function extensionality and propositional
extensionality, conclude that is-subsingleton X ＝ (X ↪ 𝟙).
Exercise. Show that the map Fin : ℕ → 𝓤₀ defined
above is left-cancellable but not
an embedding.
As we have seen, a type X can be
seen as an ∞-groupoid and hence as an ∞-category, with
identifications as the arrows. Likewise
a universe 𝓤 can be seen as the ∞-generalization of the category of
sets, with functions as the arrows. Hence a type family
A : X → 𝓤
can be seen as an ∞-presheaf, because groupoids are self-dual
categories.
With this view, the identity type former Id X : X → X → 𝓤 plays the role
of the Yoneda embedding:
𝓨 : {X : 𝓤 ̇ } → X → (X → 𝓤 ̇ ) 𝓨 {𝓤} {X} = Id X
Sometimes we want to make one of the parameters explicit:
𝑌 : (X : 𝓤 ̇ ) → X → (X → 𝓤 ̇ ) 𝑌 {𝓤} X = 𝓨 {𝓤} {X}
By our definition of Nat, for any
A : X → 𝓥 ̇ and x : X we have
Nat (𝓨 x) A = (y : X) → x ＝ y → A y,
and, by Nats-are-natural, we
have that Nat (𝓨 x) A is the type of natural transformations from
the presheaf 𝓨 x to the presheaf A. The starting point of the
Yoneda Lemma, in our context, is that every such natural
transformation is a transport.
transport-lemma : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → (τ : Nat (𝓨 x) A) → (y : X) (p : x ＝ y) → τ y p ＝ transport A p (τ x (refl x)) transport-lemma A x τ x (refl x) = refl (τ x (refl x))
We denote the point τ x (refl x) in the above lemma by 𝓔 A x τ as
refer to it as the Yoneda element of the transformation τ.
𝓔 : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → Nat (𝓨 x) A → A x 𝓔 A x τ = τ x (refl x)
The function
𝓔 A x : Nat (𝓨 x) A → A x
is an equivalence with inverse
𝓝 A x : A x → Nat (𝓨 x) A,
the transport natural transformation induced by A and x:
𝓝 : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → A x → Nat (𝓨 x) A 𝓝 A x a y p = transport A p a yoneda-η : dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext 𝓤 𝓥 → {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → 𝓝 A x ∘ 𝓔 A x ∼ id yoneda-η fe fe' A x = γ where γ : (τ : Nat (𝓨 x) A) → (λ y p → transport A p (τ x (refl x))) ＝ τ γ τ = fe (λ y → fe' (λ p → (transport-lemma A x τ y p)⁻¹)) yoneda-ε : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → 𝓔 A x ∘ 𝓝 A x ∼ id yoneda-ε A x = γ where γ : (a : A x) → transport A (refl x) a ＝ a γ = refl
By a fiberwise equivalence we mean a natural transformation whose components are all equivalences:
is-fiberwise-equiv : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → Nat A B → 𝓤 ⊔ 𝓥 ⊔ 𝓦 ̇ is-fiberwise-equiv τ = ∀ x → is-equiv (τ x) 𝓔-is-equiv : dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext 𝓤 𝓥 → {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → is-fiberwise-equiv (𝓔 A) 𝓔-is-equiv fe fe' A x = invertibles-are-equivs (𝓔 A x ) (𝓝 A x , yoneda-η fe fe' A x , yoneda-ε A x) 𝓝-is-equiv : dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext 𝓤 𝓥 → {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → is-fiberwise-equiv (𝓝 A) 𝓝-is-equiv fe fe' A x = invertibles-are-equivs (𝓝 A x) (𝓔 A x , yoneda-ε A x , yoneda-η fe fe' A x)
This gives the Yoneda Lemma for
types,
which says that natural transformations from 𝓨 x to A are in
canonical bijection with elements of A x:
Yoneda-Lemma : dfunext 𝓤 (𝓤 ⊔ 𝓥) → dfunext 𝓤 𝓥 → {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → Nat (𝓨 x) A ≃ A x Yoneda-Lemma fe fe' A x = 𝓔 A x , 𝓔-is-equiv fe fe' A x
A universal element of a
presheaf
A corresponds in our context to an element of the type is-singleton
(Σ A), which can also be written ∃! A.
If the transport transformation is a fiberwise equivalence,
then A has a universal element. More generally, we have the following:
retract-universal-lemma : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → ((y : X) → A y ◁ (x ＝ y)) → ∃! A retract-universal-lemma A x ρ = i where σ : Σ A ◁ singleton-type' x σ = Σ-retract ρ i : ∃! A i = retract-of-singleton σ (singleton-types'-are-singletons (domain A) x) fiberwise-equiv-universal : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) (τ : Nat (𝓨 x) A) → is-fiberwise-equiv τ → ∃! A fiberwise-equiv-universal A x τ e = retract-universal-lemma A x ρ where ρ : ∀ y → A y ◁ (x ＝ y) ρ y = ≃-gives-▷ ((τ y) , e y)
Conversely:
universal-fiberwise-equiv : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → ∃! A → (x : X) (τ : Nat (𝓨 x) A) → is-fiberwise-equiv τ universal-fiberwise-equiv {𝓤} {𝓥} {X} A u x τ = γ where g : singleton-type' x → Σ A g = NatΣ τ e : is-equiv g e = maps-of-singletons-are-equivs g (singleton-types'-are-singletons X x) u γ : is-fiberwise-equiv τ γ = NatΣ-equiv-gives-fiberwise-equiv τ e
In particular, the induced transport transformation τ = 𝓝 A x a is a
fiberwise equivalence if and only if there is a unique x : X with A
x, which we abbreviate as ∃! A.
A corollary is the following characterization of function
extensionality, similar to the above characterization of
univalence, by taking the transformation
τ = happly f, because hfe f says that τ is a fiberwise equivalence:
hfunext→ : hfunext 𝓤 𝓥 → ((X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (f : Π A) → ∃! g ꞉ Π A , f ∼ g) hfunext→ hfe X A f = fiberwise-equiv-universal (f ∼_) f (happly f) (hfe f) →hfunext : ((X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (f : Π A) → ∃! g ꞉ Π A , f ∼ g) → hfunext 𝓤 𝓥 →hfunext φ {X} {A} f = universal-fiberwise-equiv (f ∼_) (φ X A f) f (happly f)
We also have the following general corollaries:
fiberwise-equiv-criterion : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → ((y : X) → A y ◁ (x ＝ y)) → (τ : Nat (𝓨 x) A) → is-fiberwise-equiv τ fiberwise-equiv-criterion A x ρ τ = universal-fiberwise-equiv A (retract-universal-lemma A x ρ) x τ
This says that if we have a fiberwise retraction, then any natural transformation is an equivalence. And the following says that if we have a fiberwise equivalence, then any natural transformation is a fiberwise equivalence:
fiberwise-equiv-criterion' : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → ((y : X) → (x ＝ y) ≃ A y) → (τ : Nat (𝓨 x) A) → is-fiberwise-equiv τ fiberwise-equiv-criterion' A x e = fiberwise-equiv-criterion A x (λ y → ≃-gives-▷ (e y))
A presheaf is called representable if it is pointwise equivalent to a
presheaf of the form 𝓨 x:
_≃̇_ : {X : 𝓤 ̇ } → (X → 𝓥 ̇ ) → (X → 𝓦 ̇ ) → 𝓤 ⊔ 𝓥 ⊔ 𝓦 ̇ A ≃̇ B = ∀ x → A x ≃ B x is-representable : {X : 𝓤 ̇ } → (X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ is-representable A = Σ x ꞉ domain A , 𝓨 x ≃̇ A representable-universal : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → is-representable A → ∃! A representable-universal A (x , e) = retract-universal-lemma A x (λ x → ≃-gives-▷ (e x)) universal-representable : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → ∃! A → is-representable A universal-representable {𝓤} {𝓥} {X} {A} ((x , a) , p) = x , φ where e : is-fiberwise-equiv (𝓝 A x a) e = universal-fiberwise-equiv A ((x , a) , p) x (𝓝 A x a) φ : (y : X) → (x ＝ y) ≃ A y φ y = (𝓝 A x a y , e y)
Combining retract-universal-lemma and universal-fiberwise-equiv we get the
following:
fiberwise-retractions-are-equivs : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (x : X) → (τ : Nat (𝓨 x) A) → ((y : X) → has-section (τ y)) → is-fiberwise-equiv τ fiberwise-retractions-are-equivs {𝓤} {𝓥} {X} A x τ s = γ where ρ : (y : X) → A y ◁ (x ＝ y) ρ y = τ y , s y i : ∃! A i = retract-universal-lemma A x ρ γ : is-fiberwise-equiv τ γ = universal-fiberwise-equiv A i x τ
Perhaps the following formulation is more appealing:
fiberwise-◁-gives-≃ : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (x : X) → ((y : X) → A y ◁ (x ＝ y)) → ((y : X) → A y ≃ (x ＝ y)) fiberwise-◁-gives-≃ X A x ρ = γ where f : (y : X) → (x ＝ y) → A y f y = retraction (ρ y) e : is-fiberwise-equiv f e = fiberwise-retractions-are-equivs A x f (λ y → retraction-has-section (ρ y)) γ : (y : X) → A y ≃ (x ＝ y) γ y = ≃-sym(f y , e y)
We have the following corollary:
embedding-criterion' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → ((x x' : X) → (f x ＝ f x') ◁ (x ＝ x')) → is-embedding f embedding-criterion' f ρ = embedding-criterion f (λ x → fiberwise-◁-gives-≃ (domain f) (λ - → f x ＝ f -) x (ρ x))
Exercise. It also follows that f is an embedding if and only if the map ap f {x} {x'} has a section.
To prove that 𝓨 {𝓤} {X} is an
embedding of X into X → 𝓤 for
any type X : 𝓤, we need the following two lemmas, which are
interesting in their own right:
being-fiberwise-equiv-is-subsingleton : global-dfunext → {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {B : X → 𝓦 ̇ } → (τ : Nat A B) → is-subsingleton (is-fiberwise-equiv τ) being-fiberwise-equiv-is-subsingleton fe τ = Π-is-subsingleton fe (λ y → being-equiv-is-subsingleton fe fe (τ y)) being-representable-is-subsingleton : global-dfunext → {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) → is-subsingleton (is-representable A) being-representable-is-subsingleton fe {X} A r₀ r₁ = γ where u : ∃! A u = representable-universal A r₀ i : (x : X) (τ : Nat (𝓨 x) A) → is-singleton (is-fiberwise-equiv τ) i x τ = pointed-subsingletons-are-singletons (is-fiberwise-equiv τ) (universal-fiberwise-equiv A u x τ) (being-fiberwise-equiv-is-subsingleton fe τ) ε : (x : X) → (𝓨 x ≃̇ A) ≃ A x ε x = ((y : X) → 𝓨 x y ≃ A y) ≃⟨ ΠΣ-distr-≃ ⟩ (Σ τ ꞉ Nat (𝓨 x) A , is-fiberwise-equiv τ) ≃⟨ pr₁-≃ (i x) ⟩ Nat (𝓨 x) A ≃⟨ Yoneda-Lemma fe fe A x ⟩ A x ■ δ : is-representable A ≃ Σ A δ = Σ-cong ε v : is-singleton (is-representable A) v = equiv-to-singleton δ u γ : r₀ ＝ r₁ γ = singletons-are-subsingletons (is-representable A) v r₀ r₁
With this it is almost immediate that the Yoneda map 𝑌 X is an
embedding of X into X → 𝓤:
𝓨-is-embedding : Univalence → (X : 𝓤 ̇ ) → is-embedding (𝑌 X) 𝓨-is-embedding {𝓤} ua X A = γ where hfe : global-hfunext hfe = univalence-gives-global-hfunext ua dfe : global-dfunext dfe = univalence-gives-global-dfunext ua p = λ x → (𝓨 x ＝ A) ≃⟨ i x ⟩ ((y : X) → 𝓨 x y ＝ A y) ≃⟨ ii x ⟩ ((y : X) → 𝓨 x y ≃ A y) ■ where i = λ x → (happly (𝓨 x) A , hfe (𝓨 x) A) ii = λ x → Π-cong dfe dfe (λ y → univalence-≃ (ua 𝓤) (𝓨 x y) (A y)) e : fiber 𝓨 A ≃ is-representable A e = Σ-cong p γ : is-subsingleton (fiber 𝓨 A) γ = equiv-to-subsingleton e (being-representable-is-subsingleton dfe A)
In set theory, a function is a relation between two sets that associates to every element of the first set exactly one element of the second set. We say that the relation is functional.
In type theory, on the other hand, the notion of function is taken as primitive. However, we can show that the type of functions is equivalent to the type of functional relations. This relies on univalence.
When the types under consideration are sets, the corresponding relations are truth valued. But for the equivalence between functions and functional relations to hold for arbitrary types, we need to work with type valued relations.
More generally, we have a one-to-one corresponce between dependent
functions f : (x : X) → A x and dependent type valued functional relations R : (x : X)
→ A x → 𝓥. We take the domain X and codomain A as parameters for
a submodule:
module function-graphs (ua : Univalence) {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) where hfe : global-hfunext hfe = univalence-gives-global-hfunext ua fe : global-dfunext fe = univalence-gives-global-dfunext ua
The type of dependent functions:
Function : 𝓤 ⊔ 𝓥 ̇ Function = (x : X) → A x
That of dependent relations:
Relation : 𝓤 ⊔ (𝓥 ⁺) ̇ Relation = (x : X) → A x → 𝓥 ̇
A relation R is said to be functional if for every x : X there is
a unique a : A x with R x a:
is-functional : Relation → 𝓤 ⊔ 𝓥 ̇ is-functional R = (x : X) → ∃! a ꞉ A x , R x a
Although the relation is allowed to take values in arbitrary types, its functionality condition is a truth value:
being-functional-is-subsingleton : (R : Relation) → is-subsingleton (is-functional R) being-functional-is-subsingleton R = Π-is-subsingleton fe (λ x → ∃!-is-subsingleton (R x) fe)
The type of functional relations:
Functional-Relation : 𝓤 ⊔ (𝓥 ⁺) ̇ Functional-Relation = Σ R ꞉ Relation , is-functional R
To a function f we associate the relation R defined by R x a = (f x ＝ a). Notice that R is truth valued if the type A x is a set for every x : X, by definition of set.
ρ : Function → Relation ρ f = λ x a → f x ＝ a
To show that the map ρ is an embedding we apply the Yoneda embedding and the
fact that the map NatΠ transforms fiberwise embeddings into
embeddings:
ρ-is-embedding : is-embedding ρ ρ-is-embedding = NatΠ-is-embedding hfe hfe (λ x → 𝑌 (A x)) (λ x → 𝓨-is-embedding ua (A x)) where
This relies implicitly on the following remarks:
τ : (x : X) → A x → (A x → 𝓥 ̇ ) τ x a b = a ＝ b remark₀ : τ ＝ λ x → 𝑌 (A x) remark₀ = refl _ remark₁ : ρ ＝ NatΠ τ remark₁ = refl _
The relation induced by a function is functional, of course:
ρ-is-functional : (f : Function) → is-functional (ρ f) ρ-is-functional f = σ where σ : (x : X) → ∃! a ꞉ A x , f x ＝ a σ x = singleton-types'-are-singletons (A x) (f x)
The graph map associates functional relations to functions:
Γ : Function → Functional-Relation Γ f = ρ f , ρ-is-functional f
The function Γ can be seen as the corestriction of ρ to its image.
We get a function from a functional relation by unique choice, which is just projection:
Φ : Functional-Relation → Function Φ (R , σ) = λ x → pr₁ (center (Σ a ꞉ A x , R x a) (σ x))
To show that these two constructions are mutually inverse, we again apply the Yoneda machinery, but in a different way.
Γ-is-equiv : is-equiv Γ Γ-is-equiv = invertibles-are-equivs Γ (Φ , η , ε) where η : Φ ∘ Γ ∼ id η = refl ε : Γ ∘ Φ ∼ id ε (R , σ) = a where f : Function f = Φ (R , σ) e : (x : X) → R x (f x) e x = pr₂ (center (Σ a ꞉ A x , R x a) (σ x)) τ : (x : X) → Nat (𝓨 (f x)) (R x) τ x = 𝓝 (R x) (f x) (e x) τ-is-fiberwise-equiv : (x : X) → is-fiberwise-equiv (τ x) τ-is-fiberwise-equiv x = universal-fiberwise-equiv (R x) (σ x) (f x) (τ x) d : (x : X) (a : A x) → (f x ＝ a) ≃ R x a d x a = τ x a , τ-is-fiberwise-equiv x a c : (x : X) (a : A x) → (f x ＝ a) ＝ R x a c x a = Eq→Id (ua 𝓥) _ _ (d x a) b : ρ f ＝ R b = fe (λ x → fe (c x)) a : (ρ f , ρ-is-functional f) ＝ (R , σ) a = to-subtype-＝ being-functional-is-subsingleton b
Therefore we have a bijection between functions and functional relations:
functions-amount-to-functional-relations : Function ≃ Functional-Relation functions-amount-to-functional-relations = Γ , Γ-is-equiv
Based on the previous section, we can define a partial
function
to be a relation R such that for every x : X there is at most
one a : A x with R x a. We use Πₚ for the type of dependent
partial functions and ⇀ for the type of partial functions.
Πₚ : {X : 𝓤 ̇ } → (X → 𝓥 ̇ ) → 𝓤 ⊔ (𝓥 ⁺) ̇ Πₚ {𝓤} {𝓥} {X} A = Σ R ꞉ ((x : X) → A x → 𝓥 ̇ ) , ((x : X) → is-subsingleton (Σ a ꞉ A x , R x a)) _⇀_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ (𝓥 ⁺) ̇ X ⇀ Y = Πₚ (λ (_ : X) → Y)
Exercise. Define partial function composition, both in non-dependent and dependent versions.
is-defined : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → Πₚ A → X → 𝓥 ̇ is-defined (R , σ) x = Σ a ꞉ _ , R x a being-defined-is-subsingleton : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (f : Πₚ A) (x : X) → is-subsingleton (is-defined f x) being-defined-is-subsingleton (R , σ) x = σ x
Notice that we have to write is-defined f x, and we say that f is
defined at x, or that x is in the domain of definition of f,
rather than is-defined (f x). In fact, before being able to evaluate
a partial function f at an argument x, we need to know that f is defined
at x. However, in informal discussions we will say “f x is
defined” by the usual abuse of notation and terminology.
We will write the application of a partial function f to x, under
the information i that f x is defined, as f [ x , i ].
_[_,_] : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } (f : Πₚ A) (x : X) → is-defined f x → A x (R , s) [ x , (a , r)] = a
Exercise. Define Kleene
equality of two
partial functions f g : Πₚ A by saying that for all x : X, if
whenever one of f x and g x is defined then so is the other, and
whenever they are both defined, then they are equal:
_＝ₖ_ : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → Πₚ A → Πₚ A → 𝓤 ⊔ 𝓥 ̇ f ＝ₖ g = ∀ x → (is-defined f x ⇔ is-defined g x) × ((i : is-defined f x) (j : is-defined g x) → f [ x , i ] ＝ g [ x , j ])
Show that the equality of two partial functions, in the sense of the
identity type, is equivalent to their Kleene equality. This needs
univalence. If all types A x are sets, then functional and
propositional extensionality suffice. In the general case, it is
easier, or less hard, to approach this problem using the chapter on
equality of mathematical structures.
Example. The famous μ-operator from recursion theory is a partial function.
module μ-operator (fe : dfunext 𝓤₀ 𝓤₀) where open basic-arithmetic-and-order
First we need to show that the property of being a minimal root is a truth value and that the type of minimal roots has at most one element. It is this that requires function extensionality.
being-minimal-root-is-subsingleton : (f : ℕ → ℕ) (m : ℕ) → is-subsingleton (is-minimal-root f m) being-minimal-root-is-subsingleton f m = ×-is-subsingleton (ℕ-is-set (f m) 0) (Π-is-subsingleton fe (λ _ → Π-is-subsingleton fe (λ _ → Π-is-subsingleton fe (λ _ → 𝟘-is-subsingleton)))) minimal-root-is-subsingleton : (f : ℕ → ℕ) → is-subsingleton (minimal-root f) minimal-root-is-subsingleton f (m , p , φ) (m' , p' , φ') = to-subtype-＝ (being-minimal-root-is-subsingleton f) (at-most-one-minimal-root f m m' (p , φ) (p' , φ'))
We now define μ so that if f has a root then μ f is defined,
and, conversely, if μ f is defined then it is the minimal root of f.
Most of the work has already been done in the module
basic-arithmetic-and-order.
μ : (ℕ → ℕ) ⇀ ℕ μ = is-minimal-root , minimal-root-is-subsingleton μ-property₀ : (f : ℕ → ℕ) → (Σ n ꞉ ℕ , f n ＝ 0) → is-defined μ f μ-property₀ = root-gives-minimal-root μ-property₁ : (f : ℕ → ℕ) (i : is-defined μ f) → (f (μ [ f , i ]) ＝ 0) × ((n : ℕ) → n < μ [ f , i ] → f n ≠ 0) μ-property₁ f = pr₂
Exercise. Define
is-total : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → Πₚ A → 𝓤 ⊔ 𝓥 ̇ is-total f = ∀ x → is-defined f x
Show that the type Σ f ꞉ Πₚ A , is-total f of total partial functions is equivalent to the type Π A of functions. In particular, the type Σ f ꞉ X ⇀ Y , is-total f is equivalent to the type X → Y.
Exercise.
Two other natural renderings of the notion of partial function, for X Y : 𝓤, are given by the equivalences
(X ⇀ Y) ≃ (X → 𝓛 Y)
≃ Σ D ꞉ 𝓤 ̇ , (D ↪ X) × (D → Y)
where
𝓛 Y = Σ P ꞉ 𝓤 ̇ , is-subsingleton P × (P → Y)
≃ (𝟙 ⇀ Y)
are two equivalent formulations of the type of partial elements of
Y. Generalize the universes, and generalize these alternative descriptions of the type
of partial functions to dependent partial functions, and prove them.
Universes are not cumulative on the nose in Agda, in the sense that from
X : 𝓤
we would get that
X : 𝓤⁺
or that
X : 𝓤 ⊔ 𝓥.
Instead we work with embeddings of universes into larger universes.
The following together with its induction principle should be considered as part of the universe handling of our spartan Martin-Löf type theory:
record Lift {𝓤 : Universe} (𝓥 : Universe) (X : 𝓤 ̇ ) : 𝓤 ⊔ 𝓥 ̇ where constructor lift field lower : X open Lift public
The functions Lift, lift and lower have the following types:
type-of-Lift : type-of (Lift {𝓤} 𝓥) ＝ (𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ ) type-of-lift : {X : 𝓤 ̇ } → type-of (lift {𝓤} {𝓥} {X}) ＝ (X → Lift 𝓥 X) type-of-lower : {X : 𝓤 ̇ } → type-of (lower {𝓤} {𝓥} {X}) ＝ (Lift 𝓥 X → X) type-of-Lift = refl _ type-of-lift = refl _ type-of-lower = refl _
The induction and recursion principles are as follows:
Lift-induction : ∀ {𝓤} 𝓥 (X : 𝓤 ̇ ) (A : Lift 𝓥 X → 𝓦 ̇ ) → ((x : X) → A (lift x)) → (l : Lift 𝓥 X) → A l Lift-induction 𝓥 X A φ (lift x) = φ x Lift-recursion : ∀ {𝓤} 𝓥 {X : 𝓤 ̇ } {B : 𝓦 ̇ } → (X → B) → Lift 𝓥 X → B Lift-recursion 𝓥 {X} {B} = Lift-induction 𝓥 X (λ _ → B)
This gives an equivalence lift : X → Lift 𝓥 X and hence an embedding
Lift 𝓥 : 𝓤 ̇ → 𝓤 ⊔ 𝓥. The following two constructions can be
performed with induction, but actually hold on the nose by the so-called η rule
for
records:
lower-lift : {X : 𝓤 ̇ } (x : X) → lower {𝓤} {𝓥} (lift x) ＝ x lower-lift = refl lift-lower : {X : 𝓤 ̇ } (l : Lift 𝓥 X) → lift (lower l) ＝ l lift-lower = refl Lift-≃ : (X : 𝓤 ̇ ) → Lift 𝓥 X ≃ X Lift-≃ {𝓤} {𝓥} X = invertibility-gives-≃ lower (lift , lift-lower , lower-lift {𝓤} {𝓥}) ≃-Lift : (X : 𝓤 ̇ ) → X ≃ Lift 𝓥 X ≃-Lift {𝓤} {𝓥} X = invertibility-gives-≃ lift (lower , lower-lift {𝓤} {𝓥} , lift-lower)
With universe lifting, we can generalize equivalence induction as follows, in several steps.
Firstly, function extensionality for a pair of universes gives function extensionality for any pair of lower universes:
lower-dfunext : ∀ 𝓦 𝓣 𝓤 𝓥 → dfunext (𝓤 ⊔ 𝓦) (𝓥 ⊔ 𝓣) → dfunext 𝓤 𝓥 lower-dfunext 𝓦 𝓣 𝓤 𝓥 fe {X} {A} {f} {g} h = p where A' : Lift 𝓦 X → 𝓥 ⊔ 𝓣 ̇ A' y = Lift 𝓣 (A (lower y)) f' g' : Π A' f' y = lift (f (lower y)) g' y = lift (g (lower y)) h' : f' ∼ g' h' y = ap lift (h (lower y)) p' : f' ＝ g' p' = fe h' p : f ＝ g p = ap (λ f' x → lower (f' (lift x))) p'
This could have been used above to remove a number of function extensionality hypotheses. But also it would have required to present the material of this section before we wanted to for learning purposes.
Secondly, a function from a universe to a higher universe is an embedding provided it maps any type to an equivalent type and the two universes are univalent:
universe-embedding-criterion : is-univalent 𝓤 → is-univalent (𝓤 ⊔ 𝓥) → (f : 𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ ) → ((X : 𝓤 ̇ ) → f X ≃ X) → is-embedding f universe-embedding-criterion {𝓤} {𝓥} ua ua' f e = embedding-criterion f γ where fe : dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) fe = univalence-gives-dfunext ua' fe₀ : dfunext 𝓤 𝓤 fe₀ = lower-dfunext 𝓥 𝓥 𝓤 𝓤 fe fe₁ : dfunext 𝓤 (𝓤 ⊔ 𝓥) fe₁ = lower-dfunext 𝓥 𝓥 𝓤 (𝓤 ⊔ 𝓥) fe γ : (X X' : 𝓤 ̇ ) → (f X ＝ f X') ≃ (X ＝ X') γ X X' = (f X ＝ f X') ≃⟨ i ⟩ (f X ≃ f X') ≃⟨ ii ⟩ (X ≃ X') ≃⟨ iii ⟩ (X ＝ X') ■ where i = univalence-≃ ua' (f X) (f X') ii = Eq-Eq-cong' fe fe fe fe fe fe₀ fe₁ fe fe₀ fe₀ fe₀ fe₀ (e X) (e X') iii = ≃-sym (univalence-≃ ua X X')
In particular, the function Lift is an embedding:
Lift-is-embedding : is-univalent 𝓤 → is-univalent (𝓤 ⊔ 𝓥) → is-embedding (Lift {𝓤} 𝓥) Lift-is-embedding {𝓤} {𝓥} ua ua' = universe-embedding-criterion {𝓤} {𝓥} ua ua' (Lift 𝓥) Lift-≃
Thirdly, we have a generalization of univalence→
from a single universe to a pair of universes. We work with two
symmetrical versions, where the second is derived from the first.
We use an anonymous
module
to assume univalence in the following couple of constructions:
module _ {𝓤 𝓥 : Universe} (ua : is-univalent 𝓥) (ua' : is-univalent (𝓤 ⊔ 𝓥)) where private fe : dfunext (𝓤 ⊔ 𝓥) (𝓤 ⊔ 𝓥) fe = univalence-gives-dfunext ua' fe₀ : dfunext 𝓥 (𝓤 ⊔ 𝓥) fe₀ = lower-dfunext 𝓤 𝓤 𝓥 (𝓤 ⊔ 𝓥) fe fe₁ : dfunext 𝓤 (𝓤 ⊔ 𝓥) fe₁ = lower-dfunext (𝓤 ⊔ 𝓥) 𝓤 𝓤 (𝓤 ⊔ 𝓥) fe fe₂ : dfunext 𝓥 𝓥 fe₂ = lower-dfunext 𝓤 𝓤 𝓥 𝓥 fe fe₃ : dfunext 𝓤 𝓤 fe₃ = lower-dfunext 𝓥 𝓥 𝓤 𝓤 fe univalence→' : (X : 𝓤 ̇ ) → is-subsingleton (Σ Y ꞉ 𝓥 ̇ , X ≃ Y) univalence→' X = s where e : (Y : 𝓥 ̇ ) → (X ≃ Y) ≃ (Lift 𝓤 Y ＝ Lift 𝓥 X) e Y = (X ≃ Y) ≃⟨ i ⟩ (Y ≃ X) ≃⟨ ii ⟩ (Lift 𝓤 Y ≃ Lift 𝓥 X) ≃⟨ iii ⟩ (Lift 𝓤 Y ＝ Lift 𝓥 X) ■ where i = ≃-Sym fe₀ fe₁ fe ii = Eq-Eq-cong' fe₁ fe fe₂ fe₁ fe fe fe fe₃ fe fe fe fe (≃-Lift Y) (≃-Lift X) iii = ≃-sym (univalence-≃ ua' (Lift 𝓤 Y) (Lift 𝓥 X)) d : (Σ Y ꞉ 𝓥 ̇ , X ≃ Y) ≃ (Σ Y ꞉ 𝓥 ̇ , Lift 𝓤 Y ＝ Lift 𝓥 X) d = Σ-cong e j : is-subsingleton (Σ Y ꞉ 𝓥 ̇ , Lift 𝓤 Y ＝ Lift 𝓥 X) j = Lift-is-embedding ua ua' (Lift 𝓥 X) abstract s : is-subsingleton (Σ Y ꞉ 𝓥 ̇ , X ≃ Y) s = equiv-to-subsingleton d j univalence→'-dual : (Y : 𝓤 ̇ ) → is-subsingleton (Σ X ꞉ 𝓥 ̇ , X ≃ Y) univalence→'-dual Y = equiv-to-subsingleton e i where e : (Σ X ꞉ 𝓥 ̇ , X ≃ Y) ≃ (Σ X ꞉ 𝓥 ̇ , Y ≃ X) e = Σ-cong (λ X → ≃-Sym fe₁ fe₀ fe) i : is-subsingleton (Σ X ꞉ 𝓥 ̇ , Y ≃ X) i = univalence→' Y
This is the end of the anonymous module. We are interested in these corollaries:
univalence→'' : is-univalent (𝓤 ⊔ 𝓥) → (X : 𝓤 ̇ ) → is-subsingleton (Σ Y ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) univalence→'' ua = univalence→' ua ua univalence→''-dual : is-univalent (𝓤 ⊔ 𝓥) → (Y : 𝓤 ̇ ) → is-subsingleton (Σ X ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) univalence→''-dual ua = univalence→'-dual ua ua
The first one is applied to get the following, where Y lives in a
universe above that of X:
𝔾↑-≃ : is-univalent (𝓤 ⊔ 𝓥) → (X : 𝓤 ̇ ) (A : (Σ Y ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) → 𝓦 ̇ ) → A (Lift 𝓥 X , ≃-Lift X) → (Y : 𝓤 ⊔ 𝓥 ̇ ) (e : X ≃ Y) → A (Y , e) 𝔾↑-≃ {𝓤} {𝓥} ua X A a Y e = transport A p a where t : Σ Y ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y t = (Lift 𝓥 X , ≃-Lift X) p : t ＝ (Y , e) p = univalence→'' {𝓤} {𝓥} ua X t (Y , e) ℍ↑-≃ : is-univalent (𝓤 ⊔ 𝓥) → (X : 𝓤 ̇ ) (A : (Y : 𝓤 ⊔ 𝓥 ̇ ) → X ≃ Y → 𝓦 ̇ ) → A (Lift 𝓥 X) (≃-Lift X) → (Y : 𝓤 ⊔ 𝓥 ̇ ) (e : X ≃ Y) → A Y e ℍ↑-≃ ua X A = 𝔾↑-≃ ua X (Σ-induction A)
Exercise. Formulate and prove the equations for 𝔾↑-≃ and ℍ↑-≃
corresponding to those for 𝔾-≃ and ℍ-≃.
The difference with ℍ-≃ is that here, to get
the conclusion, we need to assume
A (Lift 𝓥 X) (≃-Lift X)
rather than
A X (id-≃).
And we have a similar development with a similar example:
𝕁↑-≃ : is-univalent (𝓤 ⊔ 𝓥) → (A : (X : 𝓤 ̇ ) (Y : 𝓤 ⊔ 𝓥 ̇ ) → X ≃ Y → 𝓦 ̇ ) → ((X : 𝓤 ̇ ) → A X (Lift 𝓥 X) (≃-Lift X)) → (X : 𝓤 ̇ ) (Y : 𝓤 ⊔ 𝓥 ̇ ) (e : X ≃ Y) → A X Y e 𝕁↑-≃ ua A φ X = ℍ↑-≃ ua X (A X) (φ X) ℍ↑-equiv : is-univalent (𝓤 ⊔ 𝓥) → (X : 𝓤 ̇ ) (A : (Y : 𝓤 ⊔ 𝓥 ̇ ) → (X → Y) → 𝓦 ̇ ) → A (Lift 𝓥 X) lift → (Y : 𝓤 ⊔ 𝓥 ̇ ) (f : X → Y) → is-equiv f → A Y f ℍ↑-equiv {𝓤} {𝓥} {𝓦} ua X A a Y f i = γ (f , i) where B : (Y : 𝓤 ⊔ 𝓥 ̇ ) → X ≃ Y → 𝓦 ̇ B Y (f , i) = A Y f b : B (Lift 𝓥 X) (≃-Lift X) b = a γ : (e : X ≃ Y) → B Y e γ = ℍ↑-≃ ua X B b Y 𝕁↑-equiv : is-univalent (𝓤 ⊔ 𝓥) → (A : (X : 𝓤 ̇ ) (Y : 𝓤 ⊔ 𝓥 ̇ ) → (X → Y) → 𝓦 ̇ ) → ((X : 𝓤 ̇ ) → A X (Lift 𝓥 X) lift) → (X : 𝓤 ̇ ) (Y : 𝓤 ⊔ 𝓥 ̇ ) (f : X → Y) → is-equiv f → A X Y f 𝕁↑-equiv ua A φ X = ℍ↑-equiv ua X (A X) (φ X)
All invertible functions from a type in a universe 𝓤 to a type in a
higher universe 𝓤 ⊔ 𝓥 satisfy a given property if (and only if) the functions
lift {𝓤} {𝓥} {X} : X → Lift 𝓥 X
satisfy the property for all X : 𝓤 (where we don’t write the
implicit arguments for lift):
𝕁↑-invertible : is-univalent (𝓤 ⊔ 𝓥) → (A : (X : 𝓤 ̇ ) (Y : 𝓤 ⊔ 𝓥 ̇ ) → (X → Y) → 𝓦 ̇ ) → ((X : 𝓤 ̇ ) → A X (Lift 𝓥 X) lift) → (X : 𝓤 ̇ ) (Y : 𝓤 ⊔ 𝓥 ̇ ) (f : X → Y) → invertible f → A X Y f 𝕁↑-invertible ua A φ X Y f i = 𝕁↑-equiv ua A φ X Y f (invertibles-are-equivs f i)
Here is an example. First, lift is a half adjoint equivalence on the nose:
lift-is-hae : (X : 𝓤 ̇ ) → is-hae {𝓤} {𝓤 ⊔ 𝓥} {X} {Lift 𝓥 X} (lift {𝓤} {𝓥}) lift-is-hae {𝓤} {𝓥} X = lower , lower-lift {𝓤} {𝓥} , lift-lower , λ x → refl (refl (lift x))
Hence all invertible maps going up universe levels are half adjoint equivalences:
equivs-are-haes↑ : is-univalent (𝓤 ⊔ 𝓥) → {X : 𝓤 ̇ } {Y : 𝓤 ⊔ 𝓥 ̇ } (f : X → Y) → is-equiv f → is-hae f equivs-are-haes↑ {𝓤} {𝓥} ua {X} {Y} = 𝕁↑-equiv {𝓤} {𝓥} ua (λ X Y f → is-hae f) lift-is-hae X Y
We have a dual development with the universes going down, where we
consider lower in place of lift:
𝔾↓-≃ : is-univalent (𝓤 ⊔ 𝓥) → (Y : 𝓤 ̇ ) (A : (Σ X ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) → 𝓦 ̇ ) → A (Lift 𝓥 Y , Lift-≃ Y) → (X : 𝓤 ⊔ 𝓥 ̇ ) (e : X ≃ Y) → A (X , e) 𝔾↓-≃ {𝓤} {𝓥} ua Y A a X e = transport A p a where t : Σ X ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y t = (Lift 𝓥 Y , Lift-≃ Y) p : t ＝ (X , e) p = univalence→'-dual {𝓤} {𝓤 ⊔ 𝓥} ua ua Y t (X , e) ℍ↓-≃ : is-univalent (𝓤 ⊔ 𝓥) → (Y : 𝓤 ̇ ) (A : (X : 𝓤 ⊔ 𝓥 ̇ ) → X ≃ Y → 𝓦 ̇ ) → A (Lift 𝓥 Y) (Lift-≃ Y) → (X : 𝓤 ⊔ 𝓥 ̇ ) (e : X ≃ Y) → A X e ℍ↓-≃ ua Y A = 𝔾↓-≃ ua Y (Σ-induction A) J↓-≃ : is-univalent (𝓤 ⊔ 𝓥) → (A : (X : 𝓤 ⊔ 𝓥 ̇ ) (Y : 𝓤 ̇ ) → X ≃ Y → 𝓦 ̇ ) → ((Y : 𝓤 ̇ ) → A (Lift 𝓥 Y) Y (Lift-≃ Y)) → (X : 𝓤 ⊔ 𝓥 ̇ ) (Y : 𝓤 ̇ ) (e : X ≃ Y) → A X Y e J↓-≃ ua A φ X Y = ℍ↓-≃ ua Y (λ X → A X Y) (φ Y) X ℍ↓-equiv : is-univalent (𝓤 ⊔ 𝓥) → (Y : 𝓤 ̇ ) (A : (X : 𝓤 ⊔ 𝓥 ̇ ) → (X → Y) → 𝓦 ̇ ) → A (Lift 𝓥 Y) lower → (X : 𝓤 ⊔ 𝓥 ̇ ) (f : X → Y) → is-equiv f → A X f ℍ↓-equiv {𝓤} {𝓥} {𝓦} ua Y A a X f i = γ (f , i) where B : (X : 𝓤 ⊔ 𝓥 ̇ ) → X ≃ Y → 𝓦 ̇ B X (f , i) = A X f b : B (Lift 𝓥 Y) (Lift-≃ Y) b = a γ : (e : X ≃ Y) → B X e γ = ℍ↓-≃ ua Y B b X 𝕁↓-equiv : is-univalent (𝓤 ⊔ 𝓥) → (A : (X : 𝓤 ⊔ 𝓥 ̇ ) (Y : 𝓤 ̇ ) → (X → Y) → 𝓦 ̇ ) → ((Y : 𝓤 ̇ ) → A (Lift 𝓥 Y) Y lower) → (X : 𝓤 ⊔ 𝓥 ̇ ) (Y : 𝓤 ̇ ) (f : X → Y) → is-equiv f → A X Y f 𝕁↓-equiv ua A φ X Y = ℍ↓-equiv ua Y (λ X → A X Y) (φ Y) X
All invertible functions from a type in a universe 𝓤 ⊔ 𝓥 to a type in the
lower universe 𝓤 satisfy a given property if (and only if) the functions
lower {𝓤} {𝓥} {Y} : Lift 𝓥 Y → Y
satisfy the property for all Y : 𝓤:
𝕁↓-invertible : is-univalent (𝓤 ⊔ 𝓥) → (A : (X : 𝓤 ⊔ 𝓥 ̇ ) (Y : 𝓤 ̇ ) → (X → Y) → 𝓦 ̇ ) → ((Y : 𝓤 ̇ ) → A (Lift 𝓥 Y) Y lower) → (X : 𝓤 ⊔ 𝓥 ̇ ) (Y : 𝓤 ̇ ) (f : X → Y) → invertible f → A X Y f 𝕁↓-invertible ua A φ X Y f i = 𝕁↓-equiv ua A φ X Y f (invertibles-are-equivs f i)
And we have similar examples:
lower-is-hae : (X : 𝓤 ̇ ) → is-hae (lower {𝓤} {𝓥} {X}) lower-is-hae {𝓤} {𝓥} X = lift , lift-lower , lower-lift {𝓤} {𝓥} , (λ x → refl (refl (lower x))) equivs-are-haes↓ : is-univalent (𝓤 ⊔ 𝓥) → {X : 𝓤 ⊔ 𝓥 ̇ } {Y : 𝓤 ̇ } (f : X → Y) → is-equiv f → is-hae f equivs-are-haes↓ {𝓤} {𝓥} ua {X} {Y} = 𝕁↓-equiv {𝓤} {𝓥} ua (λ X Y f → is-hae f) lower-is-hae X Y
A crucial example of an equivalence “going down one universe” is
Id→Eq X Y. This is because the identity type X ＝ Y lives in the
successor universe 𝓤 ⁺ if X and Y live in 𝓤, whereas the
equivalence type X ≃ Y lives in the same universe as X and
Y. Hence we can apply the above function invertibles-are-haes↓ to
Id→Eq X Y to conclude that it is a half adjoint equivalence:
Id→Eq-is-hae' : is-univalent 𝓤 → is-univalent (𝓤 ⁺) → {X Y : 𝓤 ̇ } → is-hae (Id→Eq X Y) Id→Eq-is-hae' ua ua⁺ {X} {Y} = equivs-are-haes↓ ua⁺ (Id→Eq X Y) (ua X Y)
We can be parsimonious with the uses of univalence by instead using
invertibles-are-haes, which doesn’t require univalence. However, that
Id→Eq is invertible of course requires univalence.
Id→Eq-is-hae : is-univalent 𝓤 → {X Y : 𝓤 ̇ } → is-hae (Id→Eq X Y) Id→Eq-is-hae ua {X} {Y} = invertibles-are-haes (Id→Eq X Y) (equivs-are-invertible (Id→Eq X Y) (ua X Y))
The remainder of this section is not used anywhere else. Using the
universe 𝓤ω discussed above, we can consider global properties:
global-property-of-types : 𝓤ω global-property-of-types = {𝓤 : Universe} → 𝓤 ̇ → 𝓤 ̇
We have already considered a few global properties, in fact,
such as is-singleton, is-subsingleton, is-set and _is-of-hlevel n.
We may hope to have that if A is a global property of types, then,
in the presence of univalence, for any X : 𝓤 and Y : 𝓥, if A X holds
then so does A Y if X ≃ Y. However, because we have a type of universes, or
universe levels, we may define e.g. A {𝓤} X = (𝓤 ＝ 𝓤₀), which violates
this hope. To get this conclusion, we need an assumption on A. We
say that A is cumulative if it is preserved by the embedding Lift
of universes into higher universes:
cumulative : global-property-of-types → 𝓤ω cumulative A = {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) → A X ≃ A (Lift 𝓥 X)
We can prove the following:
global-≃-ap : Univalence → (A : global-property-of-types) → cumulative A → (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → X ≃ Y → A X ≃ A Y
However, the above notion of global property is very restrictive. For
example, is-inhabited defined below
is a global property of type {𝓤 : Universe} → 𝓤 ̇ → 𝓤 ⁺ ̇ . Hence we
prove something more general, where in this example we take F 𝓤 = 𝓤 ⁺.
global-≃-ap' : Univalence → (F : Universe → Universe) → (A : {𝓤 : Universe} → 𝓤 ̇ → (F 𝓤) ̇ ) → ({𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) → A X ≃ A (Lift 𝓥 X)) → (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → X ≃ Y → A X ≃ A Y global-≃-ap' {𝓤} {𝓥} ua F A φ X Y e = A X ≃⟨ φ X ⟩ A (Lift 𝓥 X) ≃⟨ Id→Eq (A (Lift 𝓥 X)) (A (Lift 𝓤 Y)) q ⟩ A (Lift 𝓤 Y) ≃⟨ ≃-sym (φ Y) ⟩ A Y ■ where d : Lift 𝓥 X ≃ Lift 𝓤 Y d = Lift 𝓥 X ≃⟨ Lift-≃ X ⟩ X ≃⟨ e ⟩ Y ≃⟨ ≃-sym (Lift-≃ Y) ⟩ Lift 𝓤 Y ■ p : Lift 𝓥 X ＝ Lift 𝓤 Y p = Eq→Id (ua (𝓤 ⊔ 𝓥)) (Lift 𝓥 X) (Lift 𝓤 Y) d q : A (Lift 𝓥 X) ＝ A (Lift 𝓤 Y) q = ap A p
The first claim follows with F = id:
global-≃-ap ua = global-≃-ap' ua id
A subtype of a type Y is a type X together with an embedding of X into Y:
Subtype : 𝓤 ̇ → 𝓤 ⁺ ̇ Subtype {𝓤} Y = Σ X ꞉ 𝓤 ̇ , X ↪ Y
The type Ω 𝓤 of subsingletons in the universe 𝓤 is the subtype
classifier of types in 𝓤, in the sense that we have a canonical
equivalence
Subtype Y ≃ (Y → Ω 𝓤)
for any type Y : 𝓤. We will derive this from something
more general. We defined embeddings to be maps whose fibers are
all subsingletons. We can replace is-subsingleton by an arbitrary
property P of — or even structure on — types.
The following generalizes the slice
constructor _/_:
_/[_]_ : (𝓤 : Universe) → (𝓤 ̇ → 𝓥 ̇ ) → 𝓤 ̇ → 𝓤 ⁺ ⊔ 𝓥 ̇ 𝓤 /[ P ] Y = Σ X ꞉ 𝓤 ̇ , Σ f ꞉ (X → Y) , ((y : Y) → P (fiber f y)) χ-special : (P : 𝓤 ̇ → 𝓥 ̇ ) (Y : 𝓤 ̇ ) → 𝓤 /[ P ] Y → (Y → Σ P) χ-special P Y (X , f , φ) y = fiber f y , φ y is-special-map-classifier : (𝓤 ̇ → 𝓥 ̇ ) → 𝓤 ⁺ ⊔ 𝓥 ̇ is-special-map-classifier {𝓤} P = (Y : 𝓤 ̇ ) → is-equiv (χ-special P Y)
If a universe is a map classifier then Σ P is the classifier of maps
with P-fibers, for any P : 𝓤 → 𝓥:
mc-gives-sc : is-map-classifier 𝓤 → (P : 𝓤 ̇ → 𝓥 ̇ ) → is-special-map-classifier P mc-gives-sc {𝓤} s P Y = γ where e = (𝓤 /[ P ] Y) ≃⟨ ≃-sym a ⟩ (Σ σ ꞉ 𝓤 / Y , ((y : Y) → P ((χ Y) σ y))) ≃⟨ ≃-sym b ⟩ (Σ A ꞉ (Y → 𝓤 ̇ ), ((y : Y) → P (A y))) ≃⟨ ≃-sym c ⟩ (Y → Σ P) ■ where a = Σ-assoc b = Σ-change-of-variable (λ A → Π (P ∘ A)) (χ Y) (s Y) c = ΠΣ-distr-≃ observation : χ-special P Y ＝ ⌜ e ⌝ observation = refl _ γ : is-equiv (χ-special P Y) γ = ⌜⌝-is-equiv e
Therefore we have the following canonical equivalence:
χ-special-is-equiv : is-univalent 𝓤 → dfunext 𝓤 (𝓤 ⁺) → (P : 𝓤 ̇ → 𝓥 ̇ ) (Y : 𝓤 ̇ ) → is-equiv (χ-special P Y) χ-special-is-equiv {𝓤} ua fe P Y = mc-gives-sc (universes-are-map-classifiers ua fe) P Y special-map-classifier : is-univalent 𝓤 → dfunext 𝓤 (𝓤 ⁺) → (P : 𝓤 ̇ → 𝓥 ̇ ) (Y : 𝓤 ̇ ) → 𝓤 /[ P ] Y ≃ (Y → Σ P) special-map-classifier {𝓤} ua fe P Y = χ-special P Y , χ-special-is-equiv ua fe P Y
In particular, considering P = is-subsingleton, we get the promised
fact that Ω is the subtype classifier:
Ω-is-subtype-classifier : Univalence → (Y : 𝓤 ̇ ) → Subtype Y ≃ (Y → Ω 𝓤) Ω-is-subtype-classifier {𝓤} ua = special-map-classifier (ua 𝓤) (univalence-gives-dfunext' (ua 𝓤) (ua (𝓤 ⁺))) is-subsingleton
It follows that the type of subtypes of Y is always a set, even if
Y is not a set:
subtypes-form-set : Univalence → (Y : 𝓤 ̇ ) → is-set (Subtype Y) subtypes-form-set {𝓤} ua Y = equiv-to-set (Ω-is-subtype-classifier ua Y) (powersets-are-sets' ua)
We now consider P = is-singleton and the type of singletons:
𝓢 : (𝓤 : Universe) → 𝓤 ⁺ ̇ 𝓢 𝓤 = Σ S ꞉ 𝓤 ̇ , is-singleton S equiv-classification : Univalence → (Y : 𝓤 ̇ ) → (Σ X ꞉ 𝓤 ̇ , X ≃ Y) ≃ (Y → 𝓢 𝓤) equiv-classification {𝓤} ua = special-map-classifier (ua 𝓤) (univalence-gives-dfunext' (ua 𝓤) (ua (𝓤 ⁺))) is-singleton
With this we can derive a fact we already know, as follows. First the type of singletons (in a universe) is itself a singleton (in the next universe):
the-singletons-form-a-singleton : propext 𝓤 → dfunext 𝓤 𝓤 → is-singleton (𝓢 𝓤) the-singletons-form-a-singleton {𝓤} pe fe = c , φ where i : is-singleton (Lift 𝓤 𝟙) i = equiv-to-singleton (Lift-≃ 𝟙) 𝟙-is-singleton c : 𝓢 𝓤 c = Lift 𝓤 𝟙 , i φ : (x : 𝓢 𝓤) → c ＝ x φ (S , s) = to-subtype-＝ (λ _ → being-singleton-is-subsingleton fe) p where p : Lift 𝓤 𝟙 ＝ S p = pe (singletons-are-subsingletons (Lift 𝓤 𝟙) i) (singletons-are-subsingletons S s) (λ _ → center S s) (λ _ → center (Lift 𝓤 𝟙) i)
What we already knew is this:
univalence→-again : Univalence → (Y : 𝓤 ̇ ) → is-singleton (Σ X ꞉ 𝓤 ̇ , X ≃ Y) univalence→-again {𝓤} ua Y = equiv-to-singleton (equiv-classification ua Y) i where i : is-singleton (Y → 𝓢 𝓤) i = univalence-gives-vvfunext' (ua 𝓤) (ua (𝓤 ⁺)) (λ y → the-singletons-form-a-singleton (univalence-gives-propext (ua 𝓤)) (univalence-gives-dfunext (ua 𝓤)))
Exercise.
(1)
Show that the retractions into Y are classified by
the type Σ A ꞉ 𝓤 ̇ , A of pointed types.
(2) After we have
defined propositional truncation and
surjections, show that the surjections into Y are classified by the
type Σ A ꞉ 𝓤 ̇ , ∥ A ∥ of inhabited types.
We now define magma equivalences and show that the type of magma equivalences is identified with the type of magma isomorphisms. In the next section, which proves a structure identity principles, we apply this to characterize magma equality and equality of other mathematical structures in terms of equivalences of underlying types.
For simplicity we assume global univalence here.
module magma-equivalences (ua : Univalence) where open magmas dfe : global-dfunext dfe = univalence-gives-global-dfunext ua hfe : global-hfunext hfe = univalence-gives-global-hfunext ua
The magma homomorphism and isomorphism conditions are subsingleton types by virtue of the fact that the underlying type of a magma is a set by definition.
being-magma-hom-is-subsingleton : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-subsingleton (is-magma-hom M N f) being-magma-hom-is-subsingleton M N f = Π-is-subsingleton dfe (λ x → Π-is-subsingleton dfe (λ y → magma-is-set N (f (x ·⟨ M ⟩ y)) (f x ·⟨ N ⟩ f y))) being-magma-iso-is-subsingleton : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-subsingleton (is-magma-iso M N f) being-magma-iso-is-subsingleton M N f (h , g , k , η , ε) (h' , g' , k' , η' , ε') = γ where p : h ＝ h' p = being-magma-hom-is-subsingleton M N f h h' q : g ＝ g' q = dfe (λ y → g y ＝⟨ (ap g (ε' y))⁻¹ ⟩ g (f (g' y)) ＝⟨ η (g' y) ⟩ g' y ∎) i : is-subsingleton (is-magma-hom N M g' × (g' ∘ f ∼ id) × (f ∘ g' ∼ id)) i = ×-is-subsingleton (being-magma-hom-is-subsingleton N M g') (×-is-subsingleton (Π-is-subsingleton dfe (λ x → magma-is-set M (g' (f x)) x)) (Π-is-subsingleton dfe (λ y → magma-is-set N (f (g' y)) y))) γ : (h , g , k , η , ε) ＝ (h' , g' , k' , η' , ε') γ = to-×-＝ (p , to-Σ-＝ (q , i _ _))
By a magma equivalence we mean an equivalence which is a magma homomorphism. This notion is again a subsingleton type.
is-magma-equiv : (M N : Magma 𝓤) → (⟨ M ⟩ → ⟨ N ⟩) → 𝓤 ̇ is-magma-equiv M N f = is-equiv f × is-magma-hom M N f being-magma-equiv-is-subsingleton : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-subsingleton (is-magma-equiv M N f) being-magma-equiv-is-subsingleton M N f = ×-is-subsingleton (being-equiv-is-subsingleton dfe dfe f) (being-magma-hom-is-subsingleton M N f)
A function is a magma isomorphism if and only if it is a magma equivalence.
magma-isos-are-magma-equivs : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-magma-iso M N f → is-magma-equiv M N f magma-isos-are-magma-equivs M N f (h , g , k , η , ε) = i , h where i : is-equiv f i = invertibles-are-equivs f (g , η , ε) magma-equivs-are-magma-isos : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-magma-equiv M N f → is-magma-iso M N f magma-equivs-are-magma-isos M N f (i , h) = h , g , k , η , ε where g : ⟨ N ⟩ → ⟨ M ⟩ g = inverse f i η : g ∘ f ∼ id η = inverses-are-retractions f i ε : f ∘ g ∼ id ε = inverses-are-sections f i k : (a b : ⟨ N ⟩) → g (a ·⟨ N ⟩ b) ＝ g a ·⟨ M ⟩ g b k a b = g (a ·⟨ N ⟩ b) ＝⟨ ap₂ (λ a b → g (a ·⟨ N ⟩ b)) ((ε a)⁻¹) ((ε b)⁻¹) ⟩ g (f (g a) ·⟨ N ⟩ f (g b)) ＝⟨ ap g ((h (g a) (g b))⁻¹) ⟩ g (f (g a ·⟨ M ⟩ g b)) ＝⟨ η (g a ·⟨ M ⟩ g b) ⟩ g a ·⟨ M ⟩ g b ∎
Because these two notions are subsingleton types, we conclude that they are equivalent.
magma-iso-charac : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-magma-iso M N f ≃ is-magma-equiv M N f magma-iso-charac M N f = logically-equivalent-subsingletons-are-equivalent (is-magma-iso M N f) (is-magma-equiv M N f) (being-magma-iso-is-subsingleton M N f) (being-magma-equiv-is-subsingleton M N f) (magma-isos-are-magma-equivs M N f , magma-equivs-are-magma-isos M N f)
And hence they are equal by univalence.
magma-iso-charac' : (M N : Magma 𝓤) (f : ⟨ M ⟩ → ⟨ N ⟩) → is-magma-iso M N f ＝ is-magma-equiv M N f magma-iso-charac' M N f = Eq→Id (ua (universe-of ⟨ M ⟩)) (is-magma-iso M N f) (is-magma-equiv M N f) (magma-iso-charac M N f)
And by function extensionality the properties of being a magma isomorphism and a magma equivalence are the same:
magma-iso-charac'' : (M N : Magma 𝓤) → is-magma-iso M N ＝ is-magma-equiv M N magma-iso-charac'' M N = dfe (magma-iso-charac' M N)
Hence the type of magma equivalences is equivalent, and therefore equal, to the type of magma isomorphisms.
_≃ₘ_ : Magma 𝓤 → Magma 𝓤 → 𝓤 ̇ M ≃ₘ N = Σ f ꞉ (⟨ M ⟩ → ⟨ N ⟩), is-magma-equiv M N f ≅ₘ-charac : (M N : Magma 𝓤) → (M ≅ₘ N) ≃ (M ≃ₘ N) ≅ₘ-charac M N = Σ-cong (magma-iso-charac M N) ≅ₘ-charac' : (M N : Magma 𝓤) → (M ≅ₘ N) ＝ (M ≃ₘ N) ≅ₘ-charac' M N = ap Σ (magma-iso-charac'' M N)
It follows from the results of this and the next section that magma equality amounts to magma isomorphism.
Independently of any choice of foundation, we regard two groups to be the same, for all mathematical purposes, if they are isomorphic. Likewise, we consider two topological spaces to be the same if they are homeomorphic, two metric spaces to be the same if they are isometric, two categories to be the same if they are equivalent, and so on.
With Voevodsky’s univalence axiom, we can prove that these notions of sameness are automatically captured by Martin-Löf’s identity type. In particular, properties of groups are automatically invariant under isomorphism, properties of topological spaces are automatically invariant under homeomorphism, properties of metric spaces are automatically invariant under isometry, properties of categories are automatically invariant under equivalence, and so on, simply because, by design, properties are invariant under the notion of equality given by the identity type. In other foundations, the lack of such automatic invariance creates practical difficulties.
A structure identity principle describes the identity type of types of mathematical structures in terms of equivalences of underlying types, relying on univalence. The first published structure identity principle, for a large class of algebraic structures, is [Coquand and Danielsson]. The HoTT book (section 9.8) has a categorical version, whose formulation is attributed to Peter Aczel.
Here we formulate and prove a variation for types equipped with structure. We consider several versions:
One for raw structures subject to no axioms, such as ∞-magmas and pointed types.
One that adds axioms to a structure, so as to e.g. get an automatic characterization of magma identifications from a characterization of ∞-magma identifications.
One that joins two kinds of structure, so as to e.g. get an automatic characterization of identifications of pointed ∞-magmas from characterizations of identifications for pointed types and for ∞-magmas.
In particular, adding axioms to pointed ∞-magmas we get monoids with an automatic characterization of their identifications.
And then adding an axiom to monoids we get groups, again with an automatic characterization of their identitifications.
We also show that while two groups are equal precisely when they are isomorphic, two subgroups of a group are equal precisely when they have the same elements, if we define a subgroup to be a subset closed under the group operations.
We also apply these ideas to characterize identifications of metric spaces, topological spaces, graphs, partially ordered sets, categories and more.
module sip where
We consider mathematical structures specified by a function
S : 𝓤 → 𝓥
and we consider types X : 𝓤 equipped with such structure s : S X,
collected in the type
Σ X ꞉ 𝓤 , S X,
which, as we have seen, can be abbreviated as
Σ S.
For example, for the type of ∞-magmas we will take 𝓥 = 𝓤 and
S X = X → X → X.
Our objective is to describe the identity type Id (Σ S) A B, in
favourable circumstances, in terms of equivalences of the underlying
types ⟨ A ⟩ and ⟨ B ⟩ of A B : Σ S.
⟨_⟩ : {S : 𝓤 ̇ → 𝓥 ̇ } → Σ S → 𝓤 ̇ ⟨ X , s ⟩ = X structure : {S : 𝓤 ̇ → 𝓥 ̇ } (A : Σ S) → S ⟨ A ⟩ structure (X , s) = s
Our favourable circumstances will be given by data
ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓦,
ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩).
The idea is that
ι describes favourable equivalences, which will be called homomorphisms, andρ then stipulates that all identity equivalences are homomorphisms.We require that any two structures on the same type making the identity equivalence a homomorphism must be identified in a canonical way:
The canonical map
s ＝ t → ι (X , s) (X , t) (id-≃ X)
defined by induction on identifications by
refl s ↦ ρ (X , s)
must be an equivalence for all X : 𝓤 and s t : S X .
This may sound a bit abstract at this point, but in practical examples of interest it is easy to fulfill these requirements, as we will illustrate soon.
We first define the canonical map:
canonical-map : {S : 𝓤 ̇ → 𝓥 ̇ } (ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓦 ̇ ) (ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩)) {X : 𝓤 ̇ } (s t : S X) → s ＝ t → ι (X , s) (X , t) (id-≃ X) canonical-map ι ρ {X} s s (refl s) = ρ (X , s)
We refer to such favourable data as a standard notion of structure and collect them in the type
SNS S 𝓦
as follows:
SNS : (𝓤 ̇ → 𝓥 ̇ ) → (𝓦 : Universe) → 𝓤 ⁺ ⊔ 𝓥 ⊔ (𝓦 ⁺) ̇ SNS {𝓤} {𝓥} S 𝓦 = Σ ι ꞉ ((A B : Σ S) → (⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓦 ̇ )) , Σ ρ ꞉ ((A : Σ S) → ι A A (id-≃ ⟨ A ⟩)) , ({X : 𝓤 ̇ } (s t : S X) → is-equiv (canonical-map ι ρ s t))
We write homomorphic for the first projection (we don’t need
names for the other two projections):
homomorphic : {S : 𝓤 ̇ → 𝓥 ̇ } → SNS S 𝓦 → (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓦 ̇ homomorphic (ι , ρ , θ) = ι
For example, when S specifies ∞-magma structure, we will have
that homomorphic σ A B (f , i) amounts to f being a magma
homomorphism.
We then collect the homomorphic equivalences of A B : Σ S, assuming
that S is a standard notion of structure witnessed by σ, in a type
A ≃[ σ ] B.
Notice that only the first component of σ, namely homomorphic σ, is
used in the definition. The other two components are used to prove properties of A ≃[ σ ] B.
_≃[_]_ : {S : 𝓤 ̇ → 𝓥 ̇ } → Σ S → SNS S 𝓦 → Σ S → 𝓤 ⊔ 𝓦 ̇ A ≃[ σ ] B = Σ f ꞉ (⟨ A ⟩ → ⟨ B ⟩) , Σ i ꞉ is-equiv f , homomorphic σ A B (f , i) Id→homEq : {S : 𝓤 ̇ → 𝓥 ̇ } (σ : SNS S 𝓦) → (A B : Σ S) → (A ＝ B) → (A ≃[ σ ] B) Id→homEq (_ , ρ , _) A A (refl A) = id , id-is-equiv ⟨ A ⟩ , ρ A
With this we are ready to prove the promised characterization of
identity on Σ S:
homomorphism-lemma : {S : 𝓤 ̇ → 𝓥 ̇ } (σ : SNS S 𝓦) (A B : Σ S) (p : ⟨ A ⟩ ＝ ⟨ B ⟩) → (transport S p (structure A) ＝ structure B) ≃ homomorphic σ A B (Id→Eq ⟨ A ⟩ ⟨ B ⟩ p) homomorphism-lemma (ι , ρ , θ) (X , s) (X , t) (refl X) = γ where γ : (s ＝ t) ≃ ι (X , s) (X , t) (id-≃ X) γ = (canonical-map ι ρ s t , θ s t) characterization-of-＝ : is-univalent 𝓤 → {S : 𝓤 ̇ → 𝓥 ̇ } (σ : SNS S 𝓦) → (A B : Σ S) → (A ＝ B) ≃ (A ≃[ σ ] B) characterization-of-＝ ua {S} σ A B = (A ＝ B) ≃⟨ i ⟩ (Σ p ꞉ ⟨ A ⟩ ＝ ⟨ B ⟩ , transport S p (structure A) ＝ structure B) ≃⟨ ii ⟩ (Σ p ꞉ ⟨ A ⟩ ＝ ⟨ B ⟩ , ι A B (Id→Eq ⟨ A ⟩ ⟨ B ⟩ p)) ≃⟨ iii ⟩ (Σ e ꞉ ⟨ A ⟩ ≃ ⟨ B ⟩ , ι A B e) ≃⟨ iv ⟩ (A ≃[ σ ] B) ■ where ι = homomorphic σ i = Σ-＝-≃ A B ii = Σ-cong (homomorphism-lemma σ A B) iii = ≃-sym (Σ-change-of-variable (ι A B) (Id→Eq ⟨ A ⟩ ⟨ B ⟩) (ua ⟨ A ⟩ ⟨ B ⟩)) iv = Σ-assoc
This equivalence is pointwise equal to Id→homEq, and hence Id→homEq
is itself an equivalence:
Id→homEq-is-equiv : (ua : is-univalent 𝓤) {S : 𝓤 ̇ → 𝓥 ̇ } (σ : SNS S 𝓦) → (A B : Σ S) → is-equiv (Id→homEq σ A B) Id→homEq-is-equiv ua {S} σ A B = γ where h : (A B : Σ S) → Id→homEq σ A B ∼ ⌜ characterization-of-＝ ua σ A B ⌝ h A A (refl A) = refl _ γ : is-equiv (Id→homEq σ A B) γ = equivs-closed-under-∼ (⌜⌝-is-equiv (characterization-of-＝ ua σ A B)) (h A B)
We conclude this submodule with the following characterization of the canonical map and of when it is an equivalence, applying Yoneda.
module _ {S : 𝓤 ̇ → 𝓥 ̇ } (ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓦 ̇ ) (ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩)) {X : 𝓤 ̇ } where canonical-map-charac : (s t : S X) (p : s ＝ t) → canonical-map ι ρ s t p ＝ transport (λ - → ι (X , s) (X , -) (id-≃ X)) p (ρ (X , s)) canonical-map-charac s = transport-lemma (λ t → ι (X , s) (X , t) (id-≃ X)) s (canonical-map ι ρ s) when-canonical-map-is-equiv : ((s t : S X) → is-equiv (canonical-map ι ρ s t)) ⇔ ((s : S X) → ∃! t ꞉ S X , ι (X , s) (X , t) (id-≃ X)) when-canonical-map-is-equiv = (λ e s → fiberwise-equiv-universal (A s) s (τ s) (e s)) , (λ φ s → universal-fiberwise-equiv (A s) (φ s) s (τ s)) where A = λ s t → ι (X , s) (X , t) (id-≃ X) τ = canonical-map ι ρ
Another criterion is the following: It is enough to have any equivalence for the canonical map to be an equivalence:
canonical-map-equiv-criterion : ((s t : S X) → (s ＝ t) ≃ ι (X , s) (X , t) (id-≃ X)) → (s t : S X) → is-equiv (canonical-map ι ρ s t) canonical-map-equiv-criterion φ s = fiberwise-equiv-criterion' (λ t → ι (X , s) (X , t) (id-≃ X)) s (φ s) (canonical-map ι ρ s)
And in fact it is enough to have any retraction for the canonical map to be an equivalence:
canonical-map-equiv-criterion' : ((s t : S X) → ι (X , s) (X , t) (id-≃ X) ◁ (s ＝ t)) → (s t : S X) → is-equiv (canonical-map ι ρ s t) canonical-map-equiv-criterion' φ s = fiberwise-equiv-criterion (λ t → ι (X , s) (X , t) (id-≃ X)) s (φ s) (canonical-map ι ρ s)
This concludes the module sip, and we now consider some examples of uses of this.
We now make precise the example outlined above:
module ∞-magma {𝓤 : Universe} where open sip ∞-magma-structure : 𝓤 ̇ → 𝓤 ̇ ∞-magma-structure X = X → X → X ∞-Magma : 𝓤 ⁺ ̇ ∞-Magma = Σ X ꞉ 𝓤 ̇ , ∞-magma-structure X sns-data : SNS ∞-magma-structure 𝓤 sns-data = (ι , ρ , θ) where ι : (A B : ∞-Magma) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ̇ ι (X , _·_) (Y , _*_) (f , _) = (λ x x' → f (x · x')) ＝ (λ x x' → f x * f x') ρ : (A : ∞-Magma) → ι A A (id-≃ ⟨ A ⟩) ρ (X , _·_) = refl _·_ h : {X : 𝓤 ̇ } {_·_ _*_ : ∞-magma-structure X} → canonical-map ι ρ _·_ _*_ ∼ 𝑖𝑑 (_·_ ＝ _*_) h (refl _·_) = refl (refl _·_) θ : {X : 𝓤 ̇ } (_·_ _*_ : ∞-magma-structure X) → is-equiv (canonical-map ι ρ _·_ _*_) θ _·_ _*_ = equivs-closed-under-∼ (id-is-equiv (_·_ ＝ _*_)) h _≅_ : ∞-Magma → ∞-Magma → 𝓤 ̇ (X , _·_) ≅ (Y , _*_) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x')) characterization-of-∞-Magma-＝ : is-univalent 𝓤 → (A B : ∞-Magma) → (A ＝ B) ≃ (A ≅ B) characterization-of-∞-Magma-＝ ua = characterization-of-＝ ua sns-data
The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence:
characterization-of-characterization-of-∞-Magma-＝ : (ua : is-univalent 𝓤) (A : ∞-Magma) → ⌜ characterization-of-∞-Magma-＝ ua A A ⌝ (refl A) ＝ (𝑖𝑑 ⟨ A ⟩ , id-is-equiv ⟨ A ⟩ , refl _) characterization-of-characterization-of-∞-Magma-＝ ua A = refl _
Next we want to account for situations in which axioms are
considered, for example that the underlying type is a set, or that the
monoid structure satisfies the unit and associativity laws. We do this
in a submodule, by reduction to the characterization of
identifications given in the module sip.
module sip-with-axioms where open sip
The first construction, given S as above, and given
subsingleton-valued axioms for types equipped with structure specified
by S, builds SNS data on S' defined by
S' X = Σ s ꞉ S X , axioms X s
from given SNS data on S.
For that purpose we first define a forgetful map Σ S' → Σ S and
an underlying-type function Σ S → 𝓤:
[_] : {S : 𝓤 ̇ → 𝓥 ̇ } {axioms : (X : 𝓤 ̇ ) → S X → 𝓦 ̇ } → (Σ X ꞉ 𝓤 ̇ , Σ s ꞉ S X , axioms X s) → Σ S [ X , s , _ ] = (X , s) ⟪_⟫ : {S : 𝓤 ̇ → 𝓥 ̇ } {axioms : (X : 𝓤 ̇ ) → S X → 𝓦 ̇ } → (Σ X ꞉ 𝓤 ̇ , Σ s ꞉ S X , axioms X s) → 𝓤 ̇ ⟪ X , _ , _ ⟫ = X
In the following construction:
For ι' and ρ' we use ι and ρ ignoring the axioms.
For θ' we need more work, but the essence of the construction is the
fact that the projection
S' X → S X
that forgets the axioms is an embedding precisely because the axioms are subsingleton-valued.
add-axioms : {S : 𝓤 ̇ → 𝓥 ̇ } (axioms : (X : 𝓤 ̇ ) → S X → 𝓦 ̇ ) → ((X : 𝓤 ̇ ) (s : S X) → is-subsingleton (axioms X s)) → SNS S 𝓣 → SNS (λ X → Σ s ꞉ S X , axioms X s) 𝓣 add-axioms {𝓤} {𝓥} {𝓦} {𝓣} {S} axioms i (ι , ρ , θ) = ι' , ρ' , θ' where S' : 𝓤 ̇ → 𝓥 ⊔ 𝓦 ̇ S' X = Σ s ꞉ S X , axioms X s ι' : (A B : Σ S') → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓣 ̇ ι' A B = ι [ A ] [ B ] ρ' : (A : Σ S') → ι' A A (id-≃ ⟨ A ⟩) ρ' A = ρ [ A ] θ' : {X : 𝓤 ̇ } (s' t' : S' X) → is-equiv (canonical-map ι' ρ' s' t') θ' {X} (s , a) (t , b) = γ where π : S' X → S X π (s , _) = s j : is-embedding π j = pr₁-is-embedding (i X) k : {s' t' : S' X} → is-equiv (ap π {s'} {t'}) k {s'} {t'} = embedding-gives-ap-is-equiv π j s' t' l : canonical-map ι' ρ' (s , a) (t , b) ∼ canonical-map ι ρ s t ∘ ap π {s , a} {t , b} l (refl (s , a)) = refl (ρ (X , s)) e : is-equiv (canonical-map ι ρ s t ∘ ap π {s , a} {t , b}) e = ∘-is-equiv (θ s t) k γ : is-equiv (canonical-map ι' ρ' (s , a) (t , b)) γ = equivs-closed-under-∼ e l
And with this we can formulate and prove what add-axioms achieves,
namely that the characterization of the identity type remains the
same, ignoring the axioms:
characterization-of-＝-with-axioms : is-univalent 𝓤 → {S : 𝓤 ̇ → 𝓥 ̇ } (σ : SNS S 𝓣) (axioms : (X : 𝓤 ̇ ) → S X → 𝓦 ̇ ) → ((X : 𝓤 ̇ ) (s : S X) → is-subsingleton (axioms X s)) → (A B : Σ X ꞉ 𝓤 ̇ , Σ s ꞉ S X , axioms X s) → (A ＝ B) ≃ ([ A ] ≃[ σ ] [ B ]) characterization-of-＝-with-axioms ua σ axioms i = characterization-of-＝ ua (add-axioms axioms i σ)
And this concludes the module sip-with-axioms. We now consider some
examples.
As discussed above, we get magmas from ∞-magmas by adding an axiom:
module magma {𝓤 : Universe} where open sip-with-axioms Magma : 𝓤 ⁺ ̇ Magma = Σ X ꞉ 𝓤 ̇ , (X → X → X) × is-set X _≅_ : Magma → Magma → 𝓤 ̇ (X , _·_ , _) ≅ (Y , _*_ , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x')) characterization-of-Magma-＝ : is-univalent 𝓤 → (A B : Magma ) → (A ＝ B) ≃ (A ≅ B) characterization-of-Magma-＝ ua = characterization-of-＝-with-axioms ua ∞-magma.sns-data (λ X s → is-set X) (λ X s → being-set-is-subsingleton (univalence-gives-dfunext ua))
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
Exercise. Characterize identifications of monoids along the above lines. It is convenient to redefine the type of monoids to an equivalent type in the above format of structure with axioms. The following development solves this exercise.
We now discuss equality of pointed types, where a pointed type is a type equipped with a designated point.
module pointed-type {𝓤 : Universe} where open sip Pointed : 𝓤 ̇ → 𝓤 ̇ Pointed X = X sns-data : SNS Pointed 𝓤 sns-data = (ι , ρ , θ) where ι : (A B : Σ Pointed) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ̇ ι (X , x₀) (Y , y₀) (f , _) = (f x₀ ＝ y₀) ρ : (A : Σ Pointed) → ι A A (id-≃ ⟨ A ⟩) ρ (X , x₀) = refl x₀ θ : {X : 𝓤 ̇ } (x₀ x₁ : Pointed X) → is-equiv (canonical-map ι ρ x₀ x₁) θ x₀ x₁ = equivs-closed-under-∼ (id-is-equiv (x₀ ＝ x₁)) h where h : canonical-map ι ρ x₀ x₁ ∼ 𝑖𝑑 (x₀ ＝ x₁) h (refl x₀) = refl (refl x₀) _≅_ : Σ Pointed → Σ Pointed → 𝓤 ̇ (X , x₀) ≅ (Y , y₀) = Σ f ꞉ (X → Y), is-equiv f × (f x₀ ＝ y₀) characterization-of-pointed-type-＝ : is-univalent 𝓤 → (A B : Σ Pointed) → (A ＝ B) ≃ (A ≅ B) characterization-of-pointed-type-＝ ua = characterization-of-＝ ua sns-data
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
We now show how to join two mathematics structures so as to obtain a characterization of the identifications of the join from the characterization of the equalities of the structures. For example, we build the characterization of identifications of pointed ∞-magmas from the characterizations of the identifications of pointed types and the characterization of the identifications of magmas. Moreover, adding axioms, we get a characterization of identifications of monoids which amounts to the characterization of identifications of pointed ∞-magmas. Further adding an axiom, we get an automatic characterization of group identifications.
module sip-join where
We begin with the following technical lemma, whose proof uses the Yoneda machinery:
technical-lemma : {X : 𝓤 ̇ } {A : X → X → 𝓥 ̇ } {Y : 𝓦 ̇ } {B : Y → Y → 𝓣 ̇ } (f : (x₀ x₁ : X) → x₀ ＝ x₁ → A x₀ x₁) (g : (y₀ y₁ : Y) → y₀ ＝ y₁ → B y₀ y₁) → ((x₀ x₁ : X) → is-equiv (f x₀ x₁)) → ((y₀ y₁ : Y) → is-equiv (g y₀ y₁)) → ((x₀ , y₀) (x₁ , y₁) : X × Y) → is-equiv (λ (p : (x₀ , y₀) ＝ (x₁ , y₁)) → f x₀ x₁ (ap pr₁ p) , g y₀ y₁ (ap pr₂ p)) technical-lemma {𝓤} {𝓥} {𝓦} {𝓣} {X} {A} {Y} {B} f g i j (x₀ , y₀) = γ where u : ∃! x₁ ꞉ X , A x₀ x₁ u = fiberwise-equiv-universal (A x₀) x₀ (f x₀) (i x₀) v : ∃! y₁ ꞉ Y , B y₀ y₁ v = fiberwise-equiv-universal (B y₀) y₀ (g y₀) (j y₀) C : X × Y → 𝓥 ⊔ 𝓣 ̇ C (x₁ , y₁) = A x₀ x₁ × B y₀ y₁ w : (∃! x₁ ꞉ X , A x₀ x₁) → (∃! y₁ ꞉ Y , B y₀ y₁) → ∃! (x₁ , y₁) ꞉ X × Y , C (x₁ , y₁) w ((x₁ , a₁) , φ) ((y₁ , b₁) , ψ) = ((x₁ , y₁) , (a₁ , b₁)) , δ where p : ∀ x y a b → (x₁ , a₁) ＝ (x , a) → (y₁ , b₁) ＝ (y , b) → (x₁ , y₁) , (a₁ , b₁) ＝ (x , y) , (a , b) p x₁ y₁ a₁ b₁ (refl (x₁ , a₁)) (refl (y₁ , b₁)) = refl ((x₁ , y₁) , (a₁ , b₁)) δ : (((x , y) , (a , b)) : Σ C) → (x₁ , y₁) , (a₁ , b₁) ＝ ((x , y) , (a , b)) δ ((x , y) , (a , b)) = p x y a b (φ (x , a)) (ψ (y , b)) τ : Nat (𝓨 (x₀ , y₀)) C τ (x₁ , y₁) p = f x₀ x₁ (ap pr₁ p) , g y₀ y₁ (ap pr₂ p) γ : is-fiberwise-equiv τ γ = universal-fiberwise-equiv C (w u v) (x₀ , y₀) τ
We consider two given mathematical structures specified by S₀ and
S₁, and work with structures specified by their combination λ X →
S₀ X × S₁ X
variable 𝓥₀ 𝓥₁ 𝓦₀ 𝓦₁ : Universe open sip ⟪_⟫ : {S₀ : 𝓤 ̇ → 𝓥₀ ̇ } {S₁ : 𝓤 ̇ → 𝓥₁ ̇ } → (Σ X ꞉ 𝓤 ̇ , S₀ X × S₁ X) → 𝓤 ̇ ⟪ X , s₀ , s₁ ⟫ = X [_]₀ : {S₀ : 𝓤 ̇ → 𝓥₀ ̇ } {S₁ : 𝓤 ̇ → 𝓥₁ ̇ } → (Σ X ꞉ 𝓤 ̇ , S₀ X × S₁ X) → Σ S₀ [ X , s₀ , s₁ ]₀ = (X , s₀) [_]₁ : {S₀ : 𝓤 ̇ → 𝓥₀ ̇ } {S₁ : 𝓤 ̇ → 𝓥₁ ̇ } → (Σ X ꞉ 𝓤 ̇ , S₀ X × S₁ X) → Σ S₁ [ X , s₀ , s₁ ]₁ = (X , s₁)
The main construction in this submodule is this:
join : {S₀ : 𝓤 ̇ → 𝓥₀ ̇ } {S₁ : 𝓤 ̇ → 𝓥₁ ̇ } → SNS S₀ 𝓦₀ → SNS S₁ 𝓦₁ → SNS (λ X → S₀ X × S₁ X) (𝓦₀ ⊔ 𝓦₁) join {𝓤} {𝓥₀} {𝓥₁} {𝓦₀} {𝓦₁} {S₀} {S₁} (ι₀ , ρ₀ , θ₀) (ι₁ , ρ₁ , θ₁) = ι , ρ , θ where S : 𝓤 ̇ → 𝓥₀ ⊔ 𝓥₁ ̇ S X = S₀ X × S₁ X ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓦₀ ⊔ 𝓦₁ ̇ ι A B e = ι₀ [ A ]₀ [ B ]₀ e × ι₁ [ A ]₁ [ B ]₁ e ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩) ρ A = (ρ₀ [ A ]₀ , ρ₁ [ A ]₁) θ : {X : 𝓤 ̇ } (s t : S X) → is-equiv (canonical-map ι ρ s t) θ {X} (s₀ , s₁) (t₀ , t₁) = γ where c : (p : s₀ , s₁ ＝ t₀ , t₁) → ι₀ (X , s₀) (X , t₀) (id-≃ X) × ι₁ (X , s₁) (X , t₁) (id-≃ X) c p = (canonical-map ι₀ ρ₀ s₀ t₀ (ap pr₁ p) , canonical-map ι₁ ρ₁ s₁ t₁ (ap pr₂ p)) i : is-equiv c i = technical-lemma (canonical-map ι₀ ρ₀) (canonical-map ι₁ ρ₁) θ₀ θ₁ (s₀ , s₁) (t₀ , t₁) e : canonical-map ι ρ (s₀ , s₁) (t₀ , t₁) ∼ c e (refl (s₀ , s₁)) = refl (ρ₀ (X , s₀) , ρ₁ (X , s₁)) γ : is-equiv (canonical-map ι ρ (s₀ , s₁) (t₀ , t₁)) γ = equivs-closed-under-∼ i e
We then can characterize the identity type of structures in the join by the following relation:
_≃⟦_,_⟧_ : {S₀ : 𝓤 ̇ → 𝓥 ̇ } {S₁ : 𝓤 ̇ → 𝓥₁ ̇ } → (Σ X ꞉ 𝓤 ̇ , S₀ X × S₁ X) → SNS S₀ 𝓦₀ → SNS S₁ 𝓦₁ → (Σ X ꞉ 𝓤 ̇ , S₀ X × S₁ X) → 𝓤 ⊔ 𝓦₀ ⊔ 𝓦₁ ̇ A ≃⟦ σ₀ , σ₁ ⟧ B = Σ f ꞉ (⟪ A ⟫ → ⟪ B ⟫) , Σ i ꞉ is-equiv f , homomorphic σ₀ [ A ]₀ [ B ]₀ (f , i) × homomorphic σ₁ [ A ]₁ [ B ]₁ (f , i)
The following is then immediate from the join construction and the general structure identity principle:
characterization-of-join-＝ : is-univalent 𝓤 → {S₀ : 𝓤 ̇ → 𝓥 ̇ } {S₁ : 𝓤 ̇ → 𝓥₁ ̇ } (σ₀ : SNS S₀ 𝓦₀) (σ₁ : SNS S₁ 𝓦₁) (A B : Σ X ꞉ 𝓤 ̇ , S₀ X × S₁ X) → (A ＝ B) ≃ (A ≃⟦ σ₀ , σ₁ ⟧ B) characterization-of-join-＝ ua σ₀ σ₁ = characterization-of-＝ ua (join σ₀ σ₁)
This concludes the submodule. Some examples of uses of this follow.
Combining pointed types with ∞-magmas we get point ∞-magmas, and from the characterization of equality of pointed types and the characterization of equality of ∞-magmas we automatically get a characterization of the equality of pointed ∞-magmas.
module pointed-∞-magma {𝓤 : Universe} where open sip-join ∞-Magma· : 𝓤 ⁺ ̇ ∞-Magma· = Σ X ꞉ 𝓤 ̇ , (X → X → X) × X _≅_ : ∞-Magma· → ∞-Magma· → 𝓤 ̇ (X , _·_ , x₀) ≅ (Y , _*_ , y₀) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x')) × (f x₀ ＝ y₀) characterization-of-pointed-magma-＝ : is-univalent 𝓤 → (A B : ∞-Magma·) → (A ＝ B) ≃ (A ≅ B) characterization-of-pointed-magma-＝ ua = characterization-of-join-＝ ua ∞-magma.sns-data pointed-type.sns-data
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
In the following example, we combine joins and addition of axioms to get a characterization of the equality of monoids as monoid isomorphism.
module monoid {𝓤 : Universe} (ua : is-univalent 𝓤) where dfe : dfunext 𝓤 𝓤 dfe = univalence-gives-dfunext ua open sip open sip-join open sip-with-axioms open monoids hiding (Monoid) monoid-structure : 𝓤 ̇ → 𝓤 ̇ monoid-structure X = (X → X → X) × X monoid-axioms : (X : 𝓤 ̇ ) → monoid-structure X → 𝓤 ̇ monoid-axioms X (_·_ , e) = is-set X × left-neutral e _·_ × right-neutral e _·_ × associative _·_ Monoid : 𝓤 ⁺ ̇ Monoid = Σ X ꞉ 𝓤 ̇ , Σ s ꞉ monoid-structure X , monoid-axioms X s monoid-axioms-subsingleton : (X : 𝓤 ̇ ) (s : monoid-structure X) → is-subsingleton (monoid-axioms X s) monoid-axioms-subsingleton X (_·_ , e) = subsingleton-criterion' γ where γ : monoid-axioms X (_·_ , e) → is-subsingleton (monoid-axioms X (_·_ , e)) γ (i , _) = ×-is-subsingleton (being-set-is-subsingleton dfe) (×-is-subsingleton (Π-is-subsingleton dfe (λ x → i (e · x) x)) (×-is-subsingleton (Π-is-subsingleton dfe (λ x → i (x · e) x)) (Π-is-subsingleton dfe (λ x → Π-is-subsingleton dfe (λ y → Π-is-subsingleton dfe (λ z → i ((x · y) · z) (x · (y · z)))))))) sns-data : SNS (λ X → Σ s ꞉ monoid-structure X , monoid-axioms X s) 𝓤 sns-data = add-axioms monoid-axioms monoid-axioms-subsingleton (join ∞-magma.sns-data pointed-type.sns-data) _≅_ : Monoid → Monoid → 𝓤 ̇ (X , (_·_ , d) , _) ≅ (Y , (_*_ , e) , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x')) × (f d ＝ e) characterization-of-monoid-＝ : (A B : Monoid) → (A ＝ B) ≃ (A ≅ B) characterization-of-monoid-＝ = characterization-of-＝ ua sns-data
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
Exercise. A bijection that preserves the monoid multiplication automatically preserves the unit. We can say that the unit is property-like structure. This is because an associative magma, or semigroup, has at most one unit.
If we alternatively define monoids as associative magmas (that is, semigroups) with the property that a unit exists, then the structure identity principle automatically shows that identitications of monoids are equivalent to bijections that preserve the multiplication, without referring to the unit. However, functions that preserve the multiplication don’t necessarily preserve the unit (exercise), that is, are not automatically monoid homomorphisms.
monoid-structure' : 𝓤 ̇ → 𝓤 ̇ monoid-structure' X = X → X → X has-unit : {X : 𝓤 ̇ } → monoid-structure' X → 𝓤 ̇ has-unit {X} _·_ = Σ e ꞉ X , left-neutral e _·_ × right-neutral e _·_
As discussed above, the difference is that now the unit is taken to be property rather than structure:
monoid-axioms' : (X : 𝓤 ̇ ) → monoid-structure' X → 𝓤 ̇ monoid-axioms' X _·_ = is-set X × has-unit _·_ × associative _·_ Monoid' : 𝓤 ⁺ ̇ Monoid' = Σ X ꞉ 𝓤 ̇ , Σ s ꞉ monoid-structure' X , monoid-axioms' X s
The equivalence of the alternative type Monoid' with the original
type Monoid is just tuple reshuffling:
to-Monoid : Monoid' → Monoid to-Monoid (X , _·_ , i , (e , l , r) , a) = (X , (_·_ , e) , (i , l , r , a)) from-Monoid : Monoid → Monoid' from-Monoid (X , (_·_ , e) , (i , l , r , a)) = (X , _·_ , i , (e , l , r) , a) to-Monoid-is-equiv : is-equiv to-Monoid to-Monoid-is-equiv = invertibles-are-equivs to-Monoid (from-Monoid , refl , refl) from-Monoid-is-equiv : is-equiv from-Monoid from-Monoid-is-equiv = invertibles-are-equivs from-Monoid (to-Monoid , refl , refl) the-two-types-of-monoids-coincide : Monoid' ≃ Monoid the-two-types-of-monoids-coincide = to-Monoid , to-Monoid-is-equiv
And because the existence of a unit is property, the alternative monoid axioms are also property:
monoid-axioms'-subsingleton : (X : 𝓤 ̇ ) (s : monoid-structure' X) → is-subsingleton (monoid-axioms' X s) monoid-axioms'-subsingleton X _·_ = subsingleton-criterion' γ where γ : monoid-axioms' X _·_ → is-subsingleton (monoid-axioms' X _·_) γ (i , _ , _) = ×-is-subsingleton (being-set-is-subsingleton dfe) (×-is-subsingleton k (Π-is-subsingleton dfe (λ x → Π-is-subsingleton dfe (λ y → Π-is-subsingleton dfe (λ z → i ((x · y) · z) (x · (y · z))))))) where j : (e : X) → is-subsingleton (left-neutral e _·_ × right-neutral e _·_) j e = ×-is-subsingleton (Π-is-subsingleton dfe (λ x → i (e · x) x)) (Π-is-subsingleton dfe (λ x → i (x · e) x)) k : is-subsingleton (has-unit _·_) k (e , l , r) (e' , l' , r') = to-subtype-＝ j p where p = e ＝⟨ (r' e)⁻¹ ⟩ (e · e') ＝⟨ l e' ⟩ e' ∎ sns-data' : SNS (λ X → Σ s ꞉ monoid-structure' X , monoid-axioms' X s) 𝓤 sns-data' = add-axioms monoid-axioms' monoid-axioms'-subsingleton ∞-magma.sns-data
As promised above, the characterization of equality doesn’t refer to preservation of the unit:
_≅'_ : Monoid' → Monoid' → 𝓤 ̇ (X , _·_ , _) ≅' (Y , _*_ , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x')) characterization-of-monoid'-＝ : (A B : Monoid') → (A ＝ B) ≃ (A ≅' B) characterization-of-monoid'-＝ = characterization-of-＝ ua sns-data'
We can define the type of semigroup isomorphisms of monoids as follows:
_≅ₛ_ : Monoid → Monoid → 𝓤 ̇ (X , (_·_ , _) , _) ≅ₛ (Y , (_*_ , _) , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x'))
We then get the following alternative characterization of monoid equality as semigroup-isomorphism, solving the above exercise, and a bit more.
2nd-characterization-of-monoid-＝ : (A B : Monoid) → (A ＝ B) ≃ A ≅ₛ B 2nd-characterization-of-monoid-＝ A B = (A ＝ B) ≃⟨ i ⟩ (from-Monoid A ＝ from-Monoid B) ≃⟨ ii ⟩ (from-Monoid A ≅' from-Monoid B) ≃⟨ iii ⟩ (A ≅ₛ B) ■ where φ : A ＝ B → from-Monoid A ＝ from-Monoid B φ = ap from-Monoid from-Monoid-is-embedding : is-embedding from-Monoid from-Monoid-is-embedding = equivs-are-embeddings from-Monoid from-Monoid-is-equiv φ-is-equiv : is-equiv φ φ-is-equiv = embedding-gives-ap-is-equiv from-Monoid from-Monoid-is-embedding A B clearly : (from-Monoid A ≅' from-Monoid B) ＝ (A ≅ₛ B) clearly = refl _ i = (φ , φ-is-equiv) ii = characterization-of-monoid'-＝ _ _ iii = Id→Eq _ _ clearly
In the absence of the requirement that the underlying type is a set, the equivalences in the characterization of equality of associative ∞-magmas not only have to be homomorphic with respect to the magma operations but also need to respect the associativity data.
module associative-∞-magma {𝓤 : Universe} (ua : is-univalent 𝓤) where fe : dfunext 𝓤 𝓤 fe = univalence-gives-dfunext ua associative : {X : 𝓤 ̇ } → (X → X → X) → 𝓤 ̇ associative _·_ = ∀ x y z → (x · y) · z ＝ x · (y · z) ∞-amagma-structure : 𝓤 ̇ → 𝓤 ̇ ∞-amagma-structure X = Σ _·_ ꞉ (X → X → X), (associative _·_) ∞-aMagma : 𝓤 ⁺ ̇ ∞-aMagma = Σ X ꞉ 𝓤 ̇ , ∞-amagma-structure X homomorphic : {X Y : 𝓤 ̇ } → (X → X → X) → (Y → Y → Y) → (X → Y) → 𝓤 ̇ homomorphic _·_ _*_ f = (λ x y → f (x · y)) ＝ (λ x y → f x * f y)
The notion of preservation of the associativity data depends not only
on the homomorphism f but also on the homomorphism data h for f:
respect-assoc : {X A : 𝓤 ̇ } (_·_ : X → X → X) (_*_ : A → A → A) → associative _·_ → associative _*_ → (f : X → A) → homomorphic _·_ _*_ f → 𝓤 ̇ respect-assoc _·_ _*_ α β f h = fα ＝ βf where l = λ x y z → f ((x · y) · z) ＝⟨ ap (λ - → - (x · y) z) h ⟩ f (x · y) * f z ＝⟨ ap (λ - → - x y * f z) h ⟩ (f x * f y) * f z ∎ r = λ x y z → f (x · (y · z)) ＝⟨ ap (λ - → - x (y · z)) h ⟩ f x * f (y · z) ＝⟨ ap (λ - → f x * - y z) h ⟩ f x * (f y * f z) ∎ fα βf : ∀ x y z → (f x * f y) * f z ＝ f x * (f y * f z) fα x y z = (l x y z)⁻¹ ∙ ap f (α x y z) ∙ r x y z βf x y z = β (f x) (f y) (f z)
The functions l and r, defined from the binary homomorphism
condition h, give the homomorphism condition for the two induced
ternary magma operations of each magma.
The following, which holds by construction, will be used implicitly:
remark : {X : 𝓤 ̇ } (_·_ : X → X → X) (α β : associative _·_ ) → respect-assoc _·_ _·_ α β id (refl _·_) ＝ ((λ x y z → refl ((x · y) · z) ∙ ap id (α x y z)) ＝ β) remark _·_ α β = refl _
The homomorphism condition ι is then defined as expected and the
reflexivity condition ρ relies on the above remark.
open sip hiding (homomorphic) sns-data : SNS ∞-amagma-structure 𝓤 sns-data = (ι , ρ , θ) where ι : (𝓧 𝓐 : ∞-aMagma) → ⟨ 𝓧 ⟩ ≃ ⟨ 𝓐 ⟩ → 𝓤 ̇ ι (X , _·_ , α) (A , _*_ , β) (f , i) = Σ h ꞉ homomorphic _·_ _*_ f , respect-assoc _·_ _*_ α β f h ρ : (𝓧 : ∞-aMagma) → ι 𝓧 𝓧 (id-≃ ⟨ 𝓧 ⟩) ρ (X , _·_ , α) = h , p where h : homomorphic _·_ _·_ id h = refl _·_ p : (λ x y z → refl ((x · y) · z) ∙ ap id (α x y z)) ＝ α p = fe (λ x → fe (λ y → fe (λ z → refl-left ∙ ap-id (α x y z))))
We prove the canonicity condition θ with the Yoneda machinery.
u : (X : 𝓤 ̇ ) → ∀ s → ∃! t ꞉ ∞-amagma-structure X , ι (X , s) (X , t) (id-≃ X) u X (_·_ , α) = c , φ where c : Σ t ꞉ ∞-amagma-structure X , ι (X , _·_ , α) (X , t) (id-≃ X) c = (_·_ , α) , ρ (X , _·_ , α) φ : (σ : Σ t ꞉ ∞-amagma-structure X , ι (X , _·_ , α) (X , t) (id-≃ X)) → c ＝ σ φ ((_·_ , β) , refl _·_ , k) = γ where a : associative _·_ a x y z = refl ((x · y) · z) ∙ ap id (α x y z) g : singleton-type' a → Σ t ꞉ ∞-amagma-structure X , ι (X , _·_ , α) (X , t) (id-≃ X) g (β , k) = (_·_ , β) , refl _·_ , k i : is-subsingleton (singleton-type' a) i = singletons-are-subsingletons _ (singleton-types'-are-singletons _ a) q : α , pr₂ (ρ (X , _·_ , α)) ＝ β , k q = i _ _ γ : c ＝ (_·_ , β) , refl _·_ , k γ = ap g q θ : {X : 𝓤 ̇ } (s t : ∞-amagma-structure X) → is-equiv (canonical-map ι ρ s t) θ {X} s = universal-fiberwise-equiv (λ t → ι (X , s) (X , t) (id-≃ X)) (u X s) s (canonical-map ι ρ s)
The promised characterization of associative ∞-magma equality then follows directly from the general structure of identity principle:
_≅_ : ∞-aMagma → ∞-aMagma → 𝓤 ̇ (X , _·_ , α) ≅ (Y , _*_ , β) = Σ f ꞉ (X → Y) , is-equiv f × (Σ h ꞉ homomorphic _·_ _*_ f , respect-assoc _·_ _*_ α β f h) characterization-of-∞-aMagma-＝ : (A B : ∞-aMagma) → (A ＝ B) ≃ (A ≅ B) characterization-of-∞-aMagma-＝ = characterization-of-＝ ua sns-data
We add an axiom to monoids to get groups, so that we get a characterization of group equality from that of monoid equality.
module group {𝓤 : Universe} (ua : is-univalent 𝓤) where hfe : hfunext 𝓤 𝓤 hfe = univalence-gives-hfunext ua open sip open sip-with-axioms open monoid {𝓤} ua hiding (sns-data ; _≅_ ; _≅'_) group-structure : 𝓤 ̇ → 𝓤 ̇ group-structure X = Σ s ꞉ monoid-structure X , monoid-axioms X s group-axiom : (X : 𝓤 ̇ ) → monoid-structure X → 𝓤 ̇ group-axiom X (_·_ , e) = (x : X) → Σ x' ꞉ X , (x · x' ＝ e) × (x' · x ＝ e) Group : 𝓤 ⁺ ̇ Group = Σ X ꞉ 𝓤 ̇ , Σ ((_·_ , e) , a) ꞉ group-structure X , group-axiom X (_·_ , e) inv-lemma : (X : 𝓤 ̇ ) (_·_ : X → X → X) (e : X) → monoid-axioms X (_·_ , e) → (x y z : X) → (y · x) ＝ e → (x · z) ＝ e → y ＝ z inv-lemma X _·_ e (s , l , r , a) x y z q p = y ＝⟨ (r y)⁻¹ ⟩ (y · e) ＝⟨ ap (y ·_) (p ⁻¹) ⟩ (y · (x · z)) ＝⟨ (a y x z)⁻¹ ⟩ ((y · x) · z) ＝⟨ ap (_· z) q ⟩ (e · z) ＝⟨ l z ⟩ z ∎ group-axiom-is-subsingleton : (X : 𝓤 ̇ ) → (s : group-structure X) → is-subsingleton (group-axiom X (pr₁ s)) group-axiom-is-subsingleton X ((_·_ , e) , (s , l , r , a)) = γ where i : (x : X) → is-subsingleton (Σ x' ꞉ X , (x · x' ＝ e) × (x' · x ＝ e)) i x (y , _ , q) (z , p , _) = u where t : y ＝ z t = inv-lemma X _·_ e (s , l , r , a) x y z q p u : (y , _ , q) ＝ (z , p , _) u = to-subtype-＝ (λ x' → ×-is-subsingleton (s (x · x') e) (s (x' · x) e)) t γ : is-subsingleton (group-axiom X (_·_ , e)) γ = Π-is-subsingleton dfe i sns-data : SNS (λ X → Σ s ꞉ group-structure X , group-axiom X (pr₁ s)) 𝓤 sns-data = add-axioms (λ X s → group-axiom X (pr₁ s)) group-axiom-is-subsingleton (monoid.sns-data ua) _≅_ : Group → Group → 𝓤 ̇ (X , ((_·_ , d) , _) , _) ≅ (Y , ((_*_ , e) , _) , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x')) × (f d ＝ e) characterization-of-group-＝ : (A B : Group) → (A ＝ B) ≃ (A ≅ B) characterization-of-group-＝ = characterization-of-＝ ua sns-data
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
Exercise. As in monoids, the preservation of the unit follows from
the preservation of the multiplication, and hence one can remove f d
＝ e from the above definition. Prove that
(A ≅ B) ≃ (A ≅' B)
and hence, by transitivity,
(A ＝ B) ≃ (A ≅' B)
where
_≅'_ : Group → Group → 𝓤 ̇ (X , ((_·_ , d) , _) , _) ≅' (Y , ((_*_ , e) , _) , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ x x' → f (x · x')) ＝ (λ x x' → f x * f x'))
We now solve this exercise and do a bit more on the way, but in a different way as we did for monoids, for the sake of variation. We first name various projections and introduce notation.
group-structure-of : (G : Group) → group-structure ⟨ G ⟩ group-structure-of (X , ((_·_ , e) , i , l , r , a) , γ) = (_·_ , e) , i , l , r , a monoid-structure-of : (G : Group) → monoid-structure ⟨ G ⟩ monoid-structure-of (X , ((_·_ , e) , i , l , r , a) , γ) = (_·_ , e) monoid-axioms-of : (G : Group) → monoid-axioms ⟨ G ⟩ (monoid-structure-of G) monoid-axioms-of (X , ((_·_ , e) , i , l , r , a) , γ) = i , l , r , a multiplication : (G : Group) → ⟨ G ⟩ → ⟨ G ⟩ → ⟨ G ⟩ multiplication (X , ((_·_ , e) , i , l , r , a) , γ) = _·_ syntax multiplication G x y = x ·⟨ G ⟩ y unit : (G : Group) → ⟨ G ⟩ unit (X , ((_·_ , e) , i , l , r , a) , γ) = e group-is-set : (G : Group) → is-set ⟨ G ⟩ group-is-set (X , ((_·_ , e) , i , l , r , a) , γ) = i unit-left : (G : Group) (x : ⟨ G ⟩) → unit G ·⟨ G ⟩ x ＝ x unit-left (X , ((_·_ , e) , i , l , r , a) , γ) x = l x unit-right : (G : Group) (x : ⟨ G ⟩) → x ·⟨ G ⟩ unit G ＝ x unit-right (X , ((_·_ , e) , i , l , r , a) , γ) x = r x assoc : (G : Group) (x y z : ⟨ G ⟩) → (x ·⟨ G ⟩ y) ·⟨ G ⟩ z ＝ x ·⟨ G ⟩ (y ·⟨ G ⟩ z) assoc (X , ((_·_ , e) , i , l , r , a) , γ) = a inv : (G : Group) → ⟨ G ⟩ → ⟨ G ⟩ inv (X , ((_·_ , e) , i , l , r , a) , γ) x = pr₁ (γ x) inv-left : (G : Group) (x : ⟨ G ⟩) → inv G x ·⟨ G ⟩ x ＝ unit G inv-left (X , ((_·_ , e) , i , l , r , a) , γ) x = pr₂ (pr₂ (γ x)) inv-right : (G : Group) (x : ⟨ G ⟩) → x ·⟨ G ⟩ inv G x ＝ unit G inv-right (X , ((_·_ , e) , i , l , r , a) , γ) x = pr₁ (pr₂ (γ x))
We now solve the exercise.
preserves-multiplication : (G H : Group) → (⟨ G ⟩ → ⟨ H ⟩) → 𝓤 ̇ preserves-multiplication G H f = (λ (x y : ⟨ G ⟩) → f (x ·⟨ G ⟩ y)) ＝ (λ (x y : ⟨ G ⟩) → f x ·⟨ H ⟩ f y) preserves-unit : (G H : Group) → (⟨ G ⟩ → ⟨ H ⟩) → 𝓤 ̇ preserves-unit G H f = f (unit G) ＝ unit H idempotent-is-unit : (G : Group) (x : ⟨ G ⟩) → x ·⟨ G ⟩ x ＝ x → x ＝ unit G idempotent-is-unit G x p = γ where x' = inv G x γ = x ＝⟨ (unit-left G x)⁻¹ ⟩ unit G ·⟨ G ⟩ x ＝⟨ (ap (λ - → - ·⟨ G ⟩ x) (inv-left G x))⁻¹ ⟩ (x' ·⟨ G ⟩ x) ·⟨ G ⟩ x ＝⟨ assoc G x' x x ⟩ x' ·⟨ G ⟩ (x ·⟨ G ⟩ x) ＝⟨ ap (λ - → x' ·⟨ G ⟩ -) p ⟩ x' ·⟨ G ⟩ x ＝⟨ inv-left G x ⟩ unit G ∎ unit-preservation-lemma : (G H : Group) (f : ⟨ G ⟩ → ⟨ H ⟩) → preserves-multiplication G H f → preserves-unit G H f unit-preservation-lemma G H f m = idempotent-is-unit H e p where e = f (unit G) p = e ·⟨ H ⟩ e ＝⟨ ap (λ - → - (unit G) (unit G)) (m ⁻¹) ⟩ f (unit G ·⟨ G ⟩ unit G) ＝⟨ ap f (unit-left G (unit G)) ⟩ e ∎
If a map preverves multiplication then it also preserves inverses:
inv-Lemma : (G : Group) (x y z : ⟨ G ⟩) → (y ·⟨ G ⟩ x) ＝ unit G → (x ·⟨ G ⟩ z) ＝ unit G → y ＝ z inv-Lemma G = inv-lemma ⟨ G ⟩ (multiplication G) (unit G) (monoid-axioms-of G) one-left-inv : (G : Group) (x x' : ⟨ G ⟩) → (x' ·⟨ G ⟩ x) ＝ unit G → x' ＝ inv G x one-left-inv G x x' p = inv-Lemma G x x' (inv G x) p (inv-right G x) one-right-inv : (G : Group) (x x' : ⟨ G ⟩) → (x ·⟨ G ⟩ x') ＝ unit G → x' ＝ inv G x one-right-inv G x x' p = (inv-Lemma G x (inv G x) x' (inv-left G x) p)⁻¹ preserves-inv : (G H : Group) → (⟨ G ⟩ → ⟨ H ⟩) → 𝓤 ̇ preserves-inv G H f = (x : ⟨ G ⟩) → f (inv G x) ＝ inv H (f x) inv-preservation-lemma : (G H : Group) (f : ⟨ G ⟩ → ⟨ H ⟩) → preserves-multiplication G H f → preserves-inv G H f inv-preservation-lemma G H f m x = γ where p = f (inv G x) ·⟨ H ⟩ f x ＝⟨ (ap (λ - → - (inv G x) x) m)⁻¹ ⟩ f (inv G x ·⟨ G ⟩ x) ＝⟨ ap f (inv-left G x) ⟩ f (unit G) ＝⟨ unit-preservation-lemma G H f m ⟩ unit H ∎ γ : f (inv G x) ＝ inv H (f x) γ = one-left-inv H (f x) (f (inv G x)) p
The usual notion of group homomorphism is that of multiplication-preserving function. But it is known that a group homomorphism in this sense is the same thing as a function that preserves both the multiplication and the unit.
is-homomorphism : (G H : Group) → (⟨ G ⟩ → ⟨ H ⟩) → 𝓤 ̇ is-homomorphism G H f = preserves-multiplication G H f × preserves-unit G H f preservation-of-mult-is-subsingleton : (G H : Group) (f : ⟨ G ⟩ → ⟨ H ⟩) → is-subsingleton (preserves-multiplication G H f) preservation-of-mult-is-subsingleton G H f = j where j : is-subsingleton (preserves-multiplication G H f) j = Π-is-set hfe (λ _ → Π-is-set hfe (λ _ → group-is-set H)) (λ (x y : ⟨ G ⟩) → f (x ·⟨ G ⟩ y)) (λ (x y : ⟨ G ⟩) → f x ·⟨ H ⟩ f y) being-homomorphism-is-subsingleton : (G H : Group) (f : ⟨ G ⟩ → ⟨ H ⟩) → is-subsingleton (is-homomorphism G H f) being-homomorphism-is-subsingleton G H f = i where i : is-subsingleton (is-homomorphism G H f) i = ×-is-subsingleton (preservation-of-mult-is-subsingleton G H f) (group-is-set H (f (unit G)) (unit H)) notions-of-homomorphism-agree : (G H : Group) (f : ⟨ G ⟩ → ⟨ H ⟩) → is-homomorphism G H f ≃ preserves-multiplication G H f notions-of-homomorphism-agree G H f = γ where α : is-homomorphism G H f → preserves-multiplication G H f α = pr₁ β : preserves-multiplication G H f → is-homomorphism G H f β m = m , unit-preservation-lemma G H f m γ : is-homomorphism G H f ≃ preserves-multiplication G H f γ = logically-equivalent-subsingletons-are-equivalent _ _ (being-homomorphism-is-subsingleton G H f) (preservation-of-mult-is-subsingleton G H f) (α , β) ≅-agreement : (G H : Group) → (G ≅ H) ≃ (G ≅' H) ≅-agreement G H = Σ-cong (λ f → Σ-cong (λ _ → notions-of-homomorphism-agree G H f))
This equivalence is that which forgets the preservation of the unit:
forget-unit-preservation : (G H : Group) → (G ≅ H) → (G ≅' H) forget-unit-preservation G H (f , e , m , _) = f , e , m NB : (G H : Group) → ⌜ ≅-agreement G H ⌝ ＝ forget-unit-preservation G H NB G H = refl _ forget-unit-preservation-is-equiv : (G H : Group) → is-equiv (forget-unit-preservation G H) forget-unit-preservation-is-equiv G H = ⌜⌝-is-equiv (≅-agreement G H)
This completes the solution of the exercise.
We now apply the above machinery to get a characterization of equality in the slice type.
module slice {𝓤 𝓥 : Universe} (R : 𝓥 ̇ ) where open sip S : 𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ S X = X → R sns-data : SNS S (𝓤 ⊔ 𝓥) sns-data = (ι , ρ , θ) where ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ⊔ 𝓥 ̇ ι (X , g) (Y , h) (f , _) = (g ＝ h ∘ f) ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩) ρ (X , g) = refl g k : {X : 𝓤 ̇ } {g h : S X} → canonical-map ι ρ g h ∼ 𝑖𝑑 (g ＝ h) k (refl g) = refl (refl g) θ : {X : 𝓤 ̇ } (g h : S X) → is-equiv (canonical-map ι ρ g h) θ g h = equivs-closed-under-∼ (id-is-equiv (g ＝ h)) k _≅_ : 𝓤 / R → 𝓤 / R → 𝓤 ⊔ 𝓥 ̇ (X , g) ≅ (Y , h) = Σ f ꞉ (X → Y), is-equiv f × (g ＝ h ∘ f) characterization-of-/-＝ : is-univalent 𝓤 → (A B : 𝓤 / R) → (A ＝ B) ≃ (A ≅ B) characterization-of-/-＝ ua = characterization-of-＝ ua sns-data
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
It is common mathematical practice to regard isomorphic groups as the same, which is a theorem in univalent mathematics, with the notion of sameness articulated by the identity type, as shown above. However, for some purposes, we may wish to consider two groups to be the same if they have the same elements. For example, in order to show that the subgroups of a group form an algebraic lattice with the finitely generated subgroups as the compact elements, it is this notion of equality that is used, with subgroup containment as the lattice order.
Asking whether two groups have the same elements in univalent mathematics doesn’t make sense unless they are subgroups of the same ambient group. In the same way that in univalent mathematics two members of the powerset are equal iff they have the same elements, two subgroups are equal if and only if they have the same elements. The existence of an isomorphism of two subgroups does not imply their equality in the type of subgroups.
The notion of subgroup can be formulated in two equivalent ways.
A subgroup is an element of the powerset of the underlying set of the group that is closed under the group operations. So the type of subgroups of a given group is embedded as a subtype of the powerset of the underlying set and hence inherits the characterization of equality from the powerset.
A subgroup of a group G is a group H together with a
homomorphic embedding H → G. With this second definition, two
subgroups H and H' are equal iff the embeddings H → G and H' → G
can be completed to a commutative triangle by a (necessarily
unique) equivalence H → H'. So the type of subgroups of a
group G with underlying set ⟨ G ⟩ : 𝓤 is embedded in the slice type 𝓤 / ⟨ G ⟩ and hence
inherits the characterization of equality from the slice type.
module subgroup (𝓤 : Universe) (ua : Univalence) where gfe : global-dfunext gfe = univalence-gives-global-dfunext ua open sip open monoid {𝓤} (ua 𝓤) hiding (sns-data ; _≅_) open group {𝓤} (ua 𝓤)
We assume an arbitrary ambient group G in the following discussion.
module ambient (G : Group) where _·_ : ⟨ G ⟩ → ⟨ G ⟩ → ⟨ G ⟩ x · y = x ·⟨ G ⟩ y infixl 42 _·_
We abbreviate “closed under the group operations” by “group-closed”:
group-closed : (⟨ G ⟩ → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ group-closed 𝓐 = 𝓐 (unit G) × ((x y : ⟨ G ⟩) → 𝓐 x → 𝓐 y → 𝓐 (x · y)) × ((x : ⟨ G ⟩) → 𝓐 x → 𝓐 (inv G x)) Subgroup : 𝓤 ⁺ ̇ Subgroup = Σ A ꞉ 𝓟 ⟨ G ⟩ , group-closed (_∈ A) ⟪_⟫ : Subgroup → 𝓟 ⟨ G ⟩ ⟪ A , u , c , ι ⟫ = A being-group-closed-subset-is-subsingleton : (A : 𝓟 ⟨ G ⟩) → is-subsingleton (group-closed (_∈ A)) being-group-closed-subset-is-subsingleton A = ×-is-subsingleton (∈-is-subsingleton A (unit G)) (×-is-subsingleton (Π-is-subsingleton dfe (λ x → Π-is-subsingleton dfe (λ y → Π-is-subsingleton dfe (λ _ → Π-is-subsingleton dfe (λ _ → ∈-is-subsingleton A (x · y)))))) (Π-is-subsingleton dfe (λ x → Π-is-subsingleton dfe (λ _ → ∈-is-subsingleton A (inv G x))))) ⟪⟫-is-embedding : is-embedding ⟪_⟫ ⟪⟫-is-embedding = pr₁-is-embedding being-group-closed-subset-is-subsingleton
Therefore equality of subgroups is equality of their underlying subsets in the powerset:
ap-⟪⟫ : (S T : Subgroup) → S ＝ T → ⟪ S ⟫ ＝ ⟪ T ⟫ ap-⟪⟫ S T = ap ⟪_⟫ ap-⟪⟫-is-equiv : (S T : Subgroup) → is-equiv (ap-⟪⟫ S T) ap-⟪⟫-is-equiv = embedding-gives-ap-is-equiv ⟪_⟫ ⟪⟫-is-embedding subgroups-form-a-set : is-set Subgroup subgroups-form-a-set S T = equiv-to-subsingleton (ap-⟪⟫ S T , ap-⟪⟫-is-equiv S T) (powersets-are-sets' ua ⟪ S ⟫ ⟪ T ⟫)
It follows that two subgroups are equal if and only if they have the same elements:
subgroup-equality : (S T : Subgroup) → (S ＝ T) ≃ ((x : ⟨ G ⟩) → (x ∈ ⟪ S ⟫) ⇔ (x ∈ ⟪ T ⟫)) subgroup-equality S T = γ where f : S ＝ T → (x : ⟨ G ⟩) → x ∈ ⟪ S ⟫ ⇔ x ∈ ⟪ T ⟫ f p x = transport (λ - → x ∈ ⟪ - ⟫) p , transport (λ - → x ∈ ⟪ - ⟫) (p ⁻¹) h : ((x : ⟨ G ⟩) → x ∈ ⟪ S ⟫ ⇔ x ∈ ⟪ T ⟫) → ⟪ S ⟫ ＝ ⟪ T ⟫ h φ = subset-extensionality' ua α β where α : ⟪ S ⟫ ⊆ ⟪ T ⟫ α x = lr-implication (φ x) β : ⟪ T ⟫ ⊆ ⟪ S ⟫ β x = rl-implication (φ x) g : ((x : ⟨ G ⟩) → x ∈ ⟪ S ⟫ ⇔ x ∈ ⟪ T ⟫) → S ＝ T g = inverse (ap-⟪⟫ S T) (ap-⟪⟫-is-equiv S T) ∘ h γ : (S ＝ T) ≃ ((x : ⟨ G ⟩) → x ∈ ⟪ S ⟫ ⇔ x ∈ ⟪ T ⟫) γ = logically-equivalent-subsingletons-are-equivalent _ _ (subgroups-form-a-set S T) (Π-is-subsingleton dfe (λ x → ×-is-subsingleton (Π-is-subsingleton dfe (λ _ → ∈-is-subsingleton ⟪ T ⟫ x)) (Π-is-subsingleton dfe (λ _ → ∈-is-subsingleton ⟪ S ⟫ x)))) (f , g)
We now show that the type of subgroups is equivalent to the following type, as an application of the subtype classifier.
Subgroup' : 𝓤 ⁺ ̇ Subgroup' = Σ H ꞉ Group , Σ h ꞉ (⟨ H ⟩ → ⟨ G ⟩) , is-embedding h × is-homomorphism H G h
It will be convenient to introduce notation for the type of group structures satisfying the group axioms:
T : 𝓤 ̇ → 𝓤 ̇ T X = Σ ((_·_ , e) , a) ꞉ group-structure X , group-axiom X (_·_ , e)
We use an anonymous module to give common assumptions for the following few lemmas:
module _ {X : 𝓤 ̇ } (h : X → ⟨ G ⟩) (h-is-embedding : is-embedding h) where private h-lc : left-cancellable h h-lc = embeddings-are-lc h h-is-embedding having-group-closed-fiber-is-subsingleton : is-subsingleton (group-closed (fiber h)) having-group-closed-fiber-is-subsingleton = being-group-closed-subset-is-subsingleton A where A : 𝓟 ⟨ G ⟩ A y = (fiber h y , h-is-embedding y) at-most-one-homomorphic-structure : is-subsingleton (Σ τ ꞉ T X , is-homomorphism (X , τ) G h) at-most-one-homomorphic-structure ((((_*_ , unitH) , maxioms) , gaxiom) , (pmult , punit)) ((((_*'_ , unitH') , maxioms') , gaxiom') , (pmult' , punit')) = γ where τ τ' : T X τ = ((_*_ , unitH) , maxioms) , gaxiom τ' = ((_*'_ , unitH') , maxioms') , gaxiom' i : is-homomorphism (X , τ) G h i = (pmult , punit) i' : is-homomorphism (X , τ') G h i' = (pmult' , punit') p : _*_ ＝ _*'_ p = gfe (λ x → gfe (λ y → h-lc (h (x * y) ＝⟨ ap (λ - → - x y) pmult ⟩ h x · h y ＝⟨ (ap (λ - → - x y) pmult')⁻¹ ⟩ h (x *' y) ∎))) q : unitH ＝ unitH' q = h-lc (h unitH ＝⟨ punit ⟩ unit G ＝⟨ punit' ⁻¹ ⟩ h unitH' ∎) r : (_*_ , unitH) ＝ (_*'_ , unitH') r = to-×-＝ (p , q) δ : τ ＝ τ' δ = to-subtype-＝ (group-axiom-is-subsingleton X) (to-subtype-＝ (monoid-axioms-subsingleton X) r) γ : (τ , i) ＝ (τ' , i') γ = to-subtype-＝ (λ τ → being-homomorphism-is-subsingleton (X , τ) G h) δ homomorphic-structure-gives-group-closed-fiber : (Σ τ ꞉ T X , is-homomorphism (X , τ) G h) → group-closed (fiber h) homomorphic-structure-gives-group-closed-fiber ((((_*_ , unitH) , maxioms) , gaxiom) , (pmult , punit)) = (unitc , mulc , invc) where H : Group H = X , ((_*_ , unitH) , maxioms) , gaxiom unitc : fiber h (unit G) unitc = unitH , punit mulc : ((x y : ⟨ G ⟩) → fiber h x → fiber h y → fiber h (x · y)) mulc x y (a , p) (b , q) = (a * b) , (h (a * b) ＝⟨ ap (λ - → - a b) pmult ⟩ h a · h b ＝⟨ ap₂ (λ - -' → - · -') p q ⟩ x · y ∎) invc : ((x : ⟨ G ⟩) → fiber h x → fiber h (inv G x)) invc x (a , p) = inv H a , (h (inv H a) ＝⟨ inv-preservation-lemma H G h pmult a ⟩ inv G (h a) ＝⟨ ap (inv G) p ⟩ inv G x ∎)
Conversely:
group-closed-fiber-gives-homomorphic-structure : group-closed (fiber h) → (Σ τ ꞉ T X , is-homomorphism (X , τ) G h) group-closed-fiber-gives-homomorphic-structure (unitc , mulc , invc) = τ , i where φ : (x : X) → fiber h (h x) φ x = (x , refl (h x)) unitH : X unitH = fiber-point unitc _*_ : X → X → X x * y = fiber-point (mulc (h x) (h y) (φ x) (φ y)) invH : X → X invH x = fiber-point (invc (h x) (φ x)) pmul : (x y : X) → h (x * y) ＝ h x · h y pmul x y = fiber-identification (mulc (h x) (h y) (φ x) (φ y)) punit : h unitH ＝ unit G punit = fiber-identification unitc pinv : (x : X) → h (invH x) ＝ inv G (h x) pinv x = fiber-identification (invc (h x) (φ x)) unitH-left : (x : X) → unitH * x ＝ x unitH-left x = h-lc (h (unitH * x) ＝⟨ pmul unitH x ⟩ h unitH · h x ＝⟨ ap (_· h x) punit ⟩ unit G · h x ＝⟨ unit-left G (h x) ⟩ h x ∎) unitH-right : (x : X) → x * unitH ＝ x unitH-right x = h-lc (h (x * unitH) ＝⟨ pmul x unitH ⟩ h x · h unitH ＝⟨ ap (h x ·_) punit ⟩ h x · unit G ＝⟨ unit-right G (h x) ⟩ h x ∎) assocH : (x y z : X) → ((x * y) * z) ＝ (x * (y * z)) assocH x y z = h-lc (h ((x * y) * z) ＝⟨ pmul (x * y) z ⟩ h (x * y) · h z ＝⟨ ap (_· h z) (pmul x y) ⟩ (h x · h y) · h z ＝⟨ assoc G (h x) (h y) (h z) ⟩ h x · (h y · h z) ＝⟨ (ap (h x ·_) (pmul y z))⁻¹ ⟩ h x · h (y * z) ＝⟨ (pmul x (y * z))⁻¹ ⟩ h (x * (y * z)) ∎) group-axiomH : (x : X) → Σ x' ꞉ X , (x * x' ＝ unitH) × (x' * x ＝ unitH) group-axiomH x = invH x , h-lc (h (x * invH x) ＝⟨ pmul x (invH x) ⟩ h x · h (invH x) ＝⟨ ap (h x ·_) (pinv x) ⟩ h x · inv G (h x) ＝⟨ inv-right G (h x) ⟩ unit G ＝⟨ punit ⁻¹ ⟩ h unitH ∎), h-lc ((h (invH x * x) ＝⟨ pmul (invH x) x ⟩ h (invH x) · h x ＝⟨ ap (_· h x) (pinv x) ⟩ inv G (h x) · h x ＝⟨ inv-left G (h x) ⟩ unit G ＝⟨ punit ⁻¹ ⟩ h unitH ∎)) j : is-set X j = subtypes-of-sets-are-sets h h-lc (group-is-set G) τ : T X τ = ((_*_ , unitH) , (j , unitH-left , unitH-right , assocH)) , group-axiomH i : is-homomorphism (X , τ) G h i = gfe (λ x → gfe (pmul x)) , punit
What is important for our purposes is that this gives an equivalence:
fiber-structure-lemma : group-closed (fiber h) ≃ (Σ τ ꞉ T X , is-homomorphism (X , τ) G h) fiber-structure-lemma = logically-equivalent-subsingletons-are-equivalent _ _ having-group-closed-fiber-is-subsingleton at-most-one-homomorphic-structure (group-closed-fiber-gives-homomorphic-structure , homomorphic-structure-gives-group-closed-fiber)
This is the end of the anonymous submodule and we can now prove the desired result. We apply the material on the subtype classifier.
characterization-of-the-type-of-subgroups : Subgroup ≃ Subgroup' characterization-of-the-type-of-subgroups = Subgroup ≃⟨ i ⟩ (Σ A ꞉ 𝓟 ⟨ G ⟩ , group-closed (_∈ A)) ≃⟨ ii ⟩ (Σ (X , h , e) ꞉ Subtype ⟨ G ⟩ , group-closed (fiber h)) ≃⟨ iii ⟩ (Σ X ꞉ 𝓤 ̇ , Σ (h , e) ꞉ X ↪ ⟨ G ⟩ , group-closed (fiber h)) ≃⟨ iv ⟩ (Σ X ꞉ 𝓤 ̇ , Σ (h , e) ꞉ X ↪ ⟨ G ⟩ , Σ τ ꞉ T X , is-homomorphism (X , τ) G h) ≃⟨ v ⟩ (Σ X ꞉ 𝓤 ̇ , Σ h ꞉ (X → ⟨ G ⟩) , Σ e ꞉ is-embedding h , Σ τ ꞉ T X , is-homomorphism (X , τ) G h) ≃⟨ vi ⟩ (Σ X ꞉ 𝓤 ̇ , Σ h ꞉ (X → ⟨ G ⟩) , Σ τ ꞉ T X , Σ e ꞉ is-embedding h , is-homomorphism (X , τ) G h) ≃⟨ vii ⟩ (Σ X ꞉ 𝓤 ̇ , Σ τ ꞉ T X , Σ h ꞉ (X → ⟨ G ⟩) , is-embedding h × is-homomorphism (X , τ) G h) ≃⟨ viii ⟩ (Σ H ꞉ Group , Σ h ꞉ (⟨ H ⟩ → ⟨ G ⟩) , is-embedding h × is-homomorphism H G h) ≃⟨ ix ⟩ Subgroup' ■ where φ : Subtype ⟨ G ⟩ → 𝓟 ⟨ G ⟩ φ = χ-special is-subsingleton ⟨ G ⟩ j : is-equiv φ j = χ-special-is-equiv (ua 𝓤) gfe is-subsingleton ⟨ G ⟩ i = Id→Eq _ _ (refl Subgroup) ii = Σ-change-of-variable (λ (A : 𝓟 ⟨ G ⟩) → group-closed (_∈ A)) φ j iii = Σ-assoc iv = Σ-cong (λ X → Σ-cong (λ (h , e) → fiber-structure-lemma h e)) v = Σ-cong (λ X → Σ-assoc) vi = Σ-cong (λ X → Σ-cong (λ h → Σ-flip)) vii = Σ-cong (λ X → Σ-flip) viii = ≃-sym Σ-assoc ix = Id→Eq _ _ (refl Subgroup')
In particular, a subgroup induces a genuine group:
induced-group : Subgroup → Group induced-group S = pr₁ (⌜ characterization-of-the-type-of-subgroups ⌝ S)
By applying the other projections, the induced group is homomorphically embedded into the ambient group.
The crucial tool to characterize equality in the alternative type of subgroups is the following embedding into the slice type.
forgetful-map : Subgroup' → 𝓤 / ⟨ G ⟩ forgetful-map ((X , _) , h , _) = (X , h)
To show that this map is an embedding, we express it as a composition of maps that are more easily seen to be embeddings.
forgetful-map-is-embedding : is-embedding forgetful-map forgetful-map-is-embedding = γ where Subtype' : 𝓤 ̇ → 𝓤 ⁺ ̇ Subtype' X = Σ (X , h) ꞉ 𝓤 / ⟨ G ⟩ , is-embedding h f₀ : Subgroup' → Subtype ⟨ G ⟩ f₀ ((X , _) , h , e , _) = (X , h , e) f₁ : Subtype ⟨ G ⟩ → Subtype' ⟨ G ⟩ f₁ (X , h , e) = ((X , h) , e) f₂ : Subtype' ⟨ G ⟩ → 𝓤 / ⟨ G ⟩ f₂ ((X , h) , e) = (X , h) by-construction : forgetful-map ＝ f₂ ∘ f₁ ∘ f₀ by-construction = refl _ f₀-lc : left-cancellable f₀ f₀-lc {(X , τ) , h , e , i} {(X , τ') , h , e , i'} (refl (X , h , e)) = δ where p : (τ , i) ＝ (τ' , i') p = at-most-one-homomorphic-structure h e (τ , i) (τ' , i') φ : (Σ τ ꞉ T X , is-homomorphism (X , τ) G h) → Subgroup' φ (τ , i) = ((X , τ) , h , e , i) δ : ((X , τ) , h , e , i) ＝ ((X , τ') , h , e , i') δ = ap φ p f₀-is-embedding : is-embedding f₀ f₀-is-embedding = lc-maps-into-sets-are-embeddings f₀ f₀-lc (subtypes-form-set ua ⟨ G ⟩) f₁-is-equiv : is-equiv f₁ f₁-is-equiv = invertibles-are-equivs f₁ ((λ ((X , h) , e) → (X , h , e)) , refl , refl) f₁-is-embedding : is-embedding f₁ f₁-is-embedding = equivs-are-embeddings f₁ f₁-is-equiv f₂-is-embedding : is-embedding f₂ f₂-is-embedding = pr₁-is-embedding (λ (X , h) → being-embedding-is-subsingleton gfe h) γ : is-embedding forgetful-map γ = ∘-embedding f₂-is-embedding (∘-embedding f₁-is-embedding f₀-is-embedding)
With this and the characterization of equality in the slice type, we get the promised characterization of equality of the alternative type of subgroups.
_＝ₛ_ : Subgroup' → Subgroup' → 𝓤 ̇ (H , h , _ ) ＝ₛ (H' , h' , _ ) = Σ f ꞉ (⟨ H ⟩ → ⟨ H' ⟩) , is-equiv f × (h ＝ h' ∘ f) subgroup'-equality : (S T : Subgroup') → (S ＝ T) ≃ (S ＝ₛ T) subgroup'-equality S T = (S ＝ T) ≃⟨ i ⟩ (forgetful-map S ＝ forgetful-map T) ≃⟨ ii ⟩ (S ＝ₛ T) ■ where open slice ⟨ G ⟩ hiding (S) i = ≃-sym (embedding-criterion-converse forgetful-map forgetful-map-is-embedding S T) ii = characterization-of-/-＝ (ua 𝓤) (forgetful-map S) (forgetful-map T)
The equivalence f in the definition of the relation ＝ₛ is unique
when it exists, because h' is an embedding and hence is
left-cancellable. Moreover, the type S ＝ₛ T has at most one element:
subgroups'-form-a-set : is-set Subgroup' subgroups'-form-a-set = equiv-to-set (≃-sym characterization-of-the-type-of-subgroups) subgroups-form-a-set ＝ₛ-is-subsingleton-valued : (S T : Subgroup') → is-subsingleton (S ＝ₛ T) ＝ₛ-is-subsingleton-valued S T = γ where i : is-subsingleton (S ＝ T) i = subgroups'-form-a-set S T γ : is-subsingleton (S ＝ₛ T) γ = equiv-to-subsingleton (≃-sym (subgroup'-equality S T)) i
Here is an alternative proof that avoids the equivalence
Subgroup ≃ Subgroup' used above to show that the alternative type of subgroups is a set:
＝ₛ-is-subsingleton-valued' : (S S' : Subgroup') → is-subsingleton (S ＝ₛ S') ＝ₛ-is-subsingleton-valued' (H , h , e , i) (H' , h' , e' , i') = γ where S = (H , h , e , i ) S' = (H' , h' , e' , i') A = Σ f ꞉ (⟨ H ⟩ → ⟨ H' ⟩) , h' ∘ f ＝ h B = Σ (f , p) ꞉ A , is-equiv f A-is-subsingleton : is-subsingleton A A-is-subsingleton = postcomp-is-embedding gfe hfe h' e' ⟨ H ⟩ h B-is-subsingleton : is-subsingleton B B-is-subsingleton = Σ-is-subsingleton A-is-subsingleton (λ (f , p) → being-equiv-is-subsingleton gfe gfe f) δ : (S ＝ₛ S') ≃ B δ = invertibility-gives-≃ α (β , η , ε) where α = λ (f , i , p) → ((f , (p ⁻¹)) , i) β = λ ((f , p) , i) → (f , i , (p ⁻¹)) η = λ (f , i , p) → ap (λ - → (f , i , -)) (⁻¹-involutive p) ε = λ ((f , p) , i) → ap (λ - → ((f , -) , i)) (⁻¹-involutive p) γ : is-subsingleton (S ＝ₛ S') γ = equiv-to-subsingleton δ B-is-subsingleton
A mathematician asked us what a formalization of Noetherian local rings would look like in univalent type theory, in particular with respect to automatic preservation of theorems about rings by ring isomorphisms. In this section we consider rings, and we consider Noetherian local rings after we discuss unspecified existence.
We consider rings without unit, called rngs, and with unit, called rings.
There are several options to apply the above techniques to accomplish this. One way would be to add structure to Abelian groups. However, in order to avoid having to discuss the preservation of the neutral element of addition by homomorphisms separately as above in the case of groups, we proceed from scratch, where the neutral element is not part of the structure but instead its existence is part of the axioms. Cf. the discussions of equality of monoids and groups.
Exercise. Proceed using the alternative approach, which should be equally easy and short (and perhaps even shorter).
We consider r(i)ngs in a universe 𝓤, and we assume univalence in their
development. We hide the notation ⟨_⟩ from the module sip because we are going to use it for the underlying Rng of a Ring:
module ring {𝓤 : Universe} (ua : Univalence) where open sip hiding (⟨_⟩) open sip-with-axioms open sip-join
We derive function extensionality from univalence:
fe : global-dfunext fe = univalence-gives-global-dfunext ua hfe : global-hfunext hfe = univalence-gives-global-hfunext ua
We take rng structure to be the product of two magma structures:
rng-structure : 𝓤 ̇ → 𝓤 ̇ rng-structure X = (X → X → X) × (X → X → X)
The axioms are the usual ones, with the additional requirement that the underlying type is a set, as opposed to an arbitrary ∞-groupoid:
rng-axioms : (R : 𝓤 ̇ ) → rng-structure R → 𝓤 ̇ rng-axioms R (_+_ , _·_) = I × II × III × IV × V × VI × VII where I = is-set R II = (x y z : R) → (x + y) + z ＝ x + (y + z) III = (x y : R) → x + y ＝ y + x IV = Σ O ꞉ R , ((x : R) → x + O ＝ x) × ((x : R) → Σ x' ꞉ R , x + x' ＝ O) V = (x y z : R) → (x · y) · z ＝ x · (y · z) VI = (x y z : R) → x · (y + z) ＝ (x · y) + (x · z) VII = (x y z : R) → (y + z) · x ＝ (y · x) + (z · x)
The type of rngs in the universe 𝓤, which lives in the universe after 𝓤:
Rng : 𝓤 ⁺ ̇ Rng = Σ R ꞉ 𝓤 ̇ , Σ s ꞉ rng-structure R , rng-axioms R s
In order to be able to apply univalence to show that the identity type
𝓡 ＝ 𝓡' of two rngs is in canonical bijection with the type 𝓡 ≅ 𝓡'
of ring isomorphisms, we need to show that the axioms constitute
property rather than data, that is, they form a subsingleton, or a
type with at most one element. The proof is a mix of algebra (to show
that an additive semigroup has at most one zero element, and at most
one additive inverse for each element) and general facts about
subsingletons (e.g. they are closed under products) and is entirely
routine.
rng-axioms-is-subsingleton : (R : 𝓤 ̇ ) (s : rng-structure R) → is-subsingleton (rng-axioms R s) rng-axioms-is-subsingleton R (_+_ , _·_) (i , ii , iii , iv-vii) = δ where A = λ (O : R) → ((x : R) → x + O ＝ x) × ((x : R) → Σ x' ꞉ R , x + x' ＝ O) IV = Σ A a : (O O' : R) → ((x : R) → x + O ＝ x) → ((x : R) → x + O' ＝ x) → O ＝ O' a O O' f f' = O ＝⟨ (f' O)⁻¹ ⟩ (O + O') ＝⟨ iii O O' ⟩ (O' + O) ＝⟨ f O' ⟩ O' ∎ b : (O : R) → is-subsingleton ((x : R) → x + O ＝ x) b O = Π-is-subsingleton fe (λ x → i (x + O) x) c : (O : R) → ((x : R) → x + O ＝ x) → (x : R) → is-subsingleton (Σ x' ꞉ R , x + x' ＝ O) c O f x (x' , p') (x'' , p'') = to-subtype-＝ (λ x' → i (x + x') O) r where r : x' ＝ x'' r = x' ＝⟨ (f x')⁻¹ ⟩ (x' + O) ＝⟨ ap (x' +_) (p'' ⁻¹) ⟩ (x' + (x + x'')) ＝⟨ (ii x' x x'')⁻¹ ⟩ ((x' + x) + x'') ＝⟨ ap (_+ x'') (iii x' x) ⟩ ((x + x') + x'') ＝⟨ ap (_+ x'') p' ⟩ (O + x'') ＝⟨ iii O x'' ⟩ (x'' + O) ＝⟨ f x'' ⟩ x'' ∎ d : (O : R) → is-subsingleton (A O) d O (f , g) = φ (f , g) where φ : is-subsingleton (A O) φ = ×-is-subsingleton (b O) (Π-is-subsingleton fe (λ x → c O f x)) IV-is-subsingleton : is-subsingleton IV IV-is-subsingleton (O , f , g) (O' , f' , g') = e where e : (O , f , g) ＝ (O' , f' , g') e = to-subtype-＝ d (a O O' f f') γ : is-subsingleton (rng-axioms R (_+_ , _·_)) γ = ×-is-subsingleton (being-set-is-subsingleton fe) (×-is-subsingleton (Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ z → i ((x + y) + z) (x + (y + z)))))) (×-is-subsingleton (Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → i (x + y) (y + x)))) (×-is-subsingleton IV-is-subsingleton (×-is-subsingleton (Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ z → i ((x · y) · z) (x · (y · z)))))) (×-is-subsingleton (Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ z → i (x · (y + z)) ((x · y) + (x · z)))))) (Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ z → i ((y + z) · x) ((y · x) + (z · x))))))))))) δ : (α : rng-axioms R (_+_ , _·_)) → (i , ii , iii , iv-vii) ＝ α δ = γ (i , ii , iii , iv-vii)
We define a rng isomorphism to be a bijection that preserves
addition and multiplication, and collect all isomorphisms of two rngs
𝓡 and 𝓡' in a type 𝓡 ≅[Rng] 𝓡':
_≅[Rng]_ : Rng → Rng → 𝓤 ̇ (R , (_+_ , _·_) , _) ≅[Rng] (R' , (_+'_ , _·'_) , _) = Σ f ꞉ (R → R') , is-equiv f × ((λ x y → f (x + y)) ＝ (λ x y → f x +' f y)) × ((λ x y → f (x · y)) ＝ (λ x y → f x ·' f y))
Then the type of rng identities is in bijection with the type of ring isomorphisms by the above general machinery:
characterization-of-rng-＝ : (𝓡 𝓡' : Rng) → (𝓡 ＝ 𝓡') ≃ (𝓡 ≅[Rng] 𝓡') characterization-of-rng-＝ = characterization-of-＝ (ua 𝓤) (add-axioms rng-axioms rng-axioms-is-subsingleton (join ∞-magma.sns-data ∞-magma.sns-data))
The underlying type of a rng:
⟨_⟩ : (𝓡 : Rng) → 𝓤 ̇ ⟨ R , _ ⟩ = R
We now add units to rngs to get rings.
ring-structure : 𝓤 ̇ → 𝓤 ̇ ring-structure X = X × rng-structure X ring-axioms : (R : 𝓤 ̇ ) → ring-structure R → 𝓤 ̇ ring-axioms R (𝟏 , _+_ , _·_) = rng-axioms R (_+_ , _·_) × VIII where VIII = (x : R) → (x · 𝟏 ＝ x) × (𝟏 · x ＝ x) ring-axioms-is-subsingleton : (R : 𝓤 ̇ ) (s : ring-structure R) → is-subsingleton (ring-axioms R s) ring-axioms-is-subsingleton R (𝟏 , _+_ , _·_) ((i , ii-vii) , viii) = γ ((i , ii-vii) , viii) where γ : is-subsingleton (ring-axioms R (𝟏 , _+_ , _·_)) γ = ×-is-subsingleton (rng-axioms-is-subsingleton R (_+_ , _·_)) (Π-is-subsingleton fe (λ x → ×-is-subsingleton (i (x · 𝟏) x) (i (𝟏 · x) x)))
The type of rings with unit:
Ring : 𝓤 ⁺ ̇ Ring = Σ R ꞉ 𝓤 ̇ , Σ s ꞉ ring-structure R , ring-axioms R s _≅[Ring]_ : Ring → Ring → 𝓤 ̇ (R , (𝟏 , _+_ , _·_) , _) ≅[Ring] (R' , (𝟏' , _+'_ , _·'_) , _) = Σ f ꞉ (R → R') , is-equiv f × (f 𝟏 ＝ 𝟏') × ((λ x y → f (x + y)) ＝ (λ x y → f x +' f y)) × ((λ x y → f (x · y)) ＝ (λ x y → f x ·' f y)) characterization-of-ring-＝ : (𝓡 𝓡' : Ring) → (𝓡 ＝ 𝓡') ≃ (𝓡 ≅[Ring] 𝓡') characterization-of-ring-＝ = characterization-of-＝ (ua 𝓤) (add-axioms ring-axioms ring-axioms-is-subsingleton (join pointed-type.sns-data (join ∞-magma.sns-data ∞-magma.sns-data)))
We now apply the above machinery to get a characterization of equality of metric spaces, graphs and ordered structures/
module generalized-metric-space {𝓤 𝓥 : Universe} (R : 𝓥 ̇ ) (axioms : (X : 𝓤 ̇ ) → (X → X → R) → 𝓤 ⊔ 𝓥 ̇ ) (axiomss : (X : 𝓤 ̇ ) (d : X → X → R) → is-subsingleton (axioms X d)) where open sip open sip-with-axioms S : 𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ S X = X → X → R sns-data : SNS S (𝓤 ⊔ 𝓥) sns-data = (ι , ρ , θ) where ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ⊔ 𝓥 ̇ ι (X , d) (Y , e) (f , _) = (d ＝ λ x x' → e (f x) (f x')) ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩) ρ (X , d) = refl d h : {X : 𝓤 ̇ } {d e : S X} → canonical-map ι ρ d e ∼ 𝑖𝑑 (d ＝ e) h (refl d) = refl (refl d) θ : {X : 𝓤 ̇ } (d e : S X) → is-equiv (canonical-map ι ρ d e) θ d e = equivs-closed-under-∼ (id-is-equiv (d ＝ e)) h M : 𝓤 ⁺ ⊔ 𝓥 ̇ M = Σ X ꞉ 𝓤 ̇ , Σ d ꞉ (X → X → R) , axioms X d _≅_ : M → M → 𝓤 ⊔ 𝓥 ̇ (X , d , _) ≅ (Y , e , _) = Σ f ꞉ (X → Y), is-equiv f × (d ＝ λ x x' → e (f x) (f x')) characterization-of-M-＝ : is-univalent 𝓤 → (A B : M) → (A ＝ B) ≃ (A ≅ B) characterization-of-M-＝ ua = characterization-of-＝-with-axioms ua sns-data axioms axiomss
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
We have the following particular cases of interest:
Metric spaces. If R is a type of real numbers, then the axioms
can be taken to be those for metric spaces, in which case M
amounts to the type of metric spaces. Then the above characterizes
metric space identification as isometry.
Graphs. If R is the type of truth values, and the axioms
function is constant with value true, then M amounts to the
type of directed graphs, and the above characterizes graph
identification as graph isomorphism. We get undirected graphs by
requiring the relation to be symmetric in the axioms.
Partially ordered sets. Again with R taken to be the type of
truth values and suitable axioms, we get posets and other ordered
structures, and the above says that their identifications amount
to order isomorphisms.
We get a type of topological spaces when R
is the type of truth values and the axioms are appropriately chosen.
module generalized-topological-space (𝓤 𝓥 : Universe) (R : 𝓥 ̇ ) (axioms : (X : 𝓤 ̇ ) → ((X → R) → R) → 𝓤 ⊔ 𝓥 ̇ ) (axiomss : (X : 𝓤 ̇ ) (𝓞 : (X → R) → R) → is-subsingleton (axioms X 𝓞)) where open sip open sip-with-axioms
When R is the type of truth values, the type (X → R) is the
powerset of X, and membership amounts to function application:
ℙ : 𝓦 ̇ → 𝓥 ⊔ 𝓦 ̇ ℙ X = X → R _∊_ : {X : 𝓦 ̇ } → X → ℙ X → R x ∊ A = A x inverse-image : {X Y : 𝓤 ̇ } → (X → Y) → ℙ Y → ℙ X inverse-image f B = λ x → f x ∊ B ℙℙ : 𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ ℙℙ X = ℙ (ℙ X) Space : 𝓤 ⁺ ⊔ 𝓥 ̇ Space = Σ X ꞉ 𝓤 ̇ , Σ 𝓞 ꞉ ℙℙ X , axioms X 𝓞
If (X , 𝓞X , a) and (Y , 𝓞Y , b) are spaces, a
homeomorphism can be
described as a bijection f : X → Y such that the open sets of Y
are precisely those whose inverse images are open in X, which can be
written as
(λ (V : ℙ Y) → inverse-image f V ∊ 𝓞X) ＝ 𝓞Y
Then ι defined below expresses the fact that a given bijection is a
homeomorphism:
sns-data : SNS ℙℙ (𝓤 ⊔ 𝓥) sns-data = (ι , ρ , θ) where ι : (A B : Σ ℙℙ) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ⊔ 𝓥 ̇ ι (X , 𝓞X) (Y , 𝓞Y) (f , _) = (λ (V : ℙ Y) → inverse-image f V ∊ 𝓞X) ＝ 𝓞Y
What ρ says is that identity function is a homeomorphism, trivially:
ρ : (A : Σ ℙℙ) → ι A A (id-≃ ⟨ A ⟩) ρ (X , 𝓞) = refl 𝓞
Then θ amounts to the fact that two topologies on the same set must
be the same if they make the identity function into a homeomorphism.
h : {X : 𝓤 ̇ } {𝓞 𝓞' : ℙℙ X} → canonical-map ι ρ 𝓞 𝓞' ∼ 𝑖𝑑 (𝓞 ＝ 𝓞') h (refl 𝓞) = refl (refl 𝓞) θ : {X : 𝓤 ̇ } (𝓞 𝓞' : ℙℙ X) → is-equiv (canonical-map ι ρ 𝓞 𝓞') θ {X} 𝓞 𝓞' = equivs-closed-under-∼ (id-is-equiv (𝓞 ＝ 𝓞')) h
We introduce notation for the type of homeomorphisms:
_≅_ : Space → Space → 𝓤 ⊔ 𝓥 ̇ (X , 𝓞X , _) ≅ (Y , 𝓞Y , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ V → inverse-image f V ∊ 𝓞X) ＝ 𝓞Y) characterization-of-Space-＝ : is-univalent 𝓤 → (A B : Space) → (A ＝ B) ≃ (A ≅ B) characterization-of-Space-＝ ua = characterization-of-＝-with-axioms ua sns-data axioms axiomss
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
But of course there are other choices for R that also make
sense. For example, we can take R to be a type of real numbers, with
the axioms for X and F : (X → R) → R saying that F is a linear
functional. Then the above gives a characterization of the identity
type of types equipped with linear functionals, in which case we may
prefer to rephrase the above as
_≅'_ : Space → Space → 𝓤 ⊔ 𝓥 ̇ (X , F , _) ≅' (Y , G , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ (v : Y → R) → F (v ∘ f)) ＝ G) characterization-of-Space-＝' : is-univalent 𝓤 → (A B : Space) → (A ＝ B) ≃ (A ≅' B) characterization-of-Space-＝' = characterization-of-Space-＝
Linear functions on certain spaces correspond to special kinds of measures by the Riesz representation theorem, and hence in this case Space becomes a type of such kind of measure spaces by an appropriate choice of axioms.
By a selection space we mean a type X equipped with a selection
function, where a selection function is a map (X → R) → X, where the
type R is a parameter.
module selection-space (𝓤 𝓥 : Universe) (R : 𝓥 ̇ ) (axioms : (X : 𝓤 ̇ ) → ((X → R) → X) → 𝓤 ⊔ 𝓥 ̇ ) (axiomss : (X : 𝓤 ̇ ) (ε : (X → R) → X) → is-subsingleton (axioms X ε)) where open sip open sip-with-axioms S : 𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ S X = (X → R) → X SelectionSpace : 𝓤 ⁺ ⊔ 𝓥 ̇ SelectionSpace = Σ X ꞉ 𝓤 ̇ , Σ ε ꞉ S X , axioms X ε sns-data : SNS S (𝓤 ⊔ 𝓥) sns-data = (ι , ρ , θ) where ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ⊔ 𝓥 ̇ ι (X , ε) (Y , δ) (f , _) = (λ (q : Y → R) → f (ε (q ∘ f))) ＝ δ ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩) ρ (X , ε) = refl ε θ : {X : 𝓤 ̇ } (ε δ : S X) → is-equiv (canonical-map ι ρ ε δ) θ {X} ε δ = γ where h : canonical-map ι ρ ε δ ∼ 𝑖𝑑 (ε ＝ δ) h (refl ε) = refl (refl ε) γ : is-equiv (canonical-map ι ρ ε δ) γ = equivs-closed-under-∼ (id-is-equiv (ε ＝ δ)) h _≅_ : SelectionSpace → SelectionSpace → 𝓤 ⊔ 𝓥 ̇ (X , ε , _) ≅ (Y , δ , _) = Σ f ꞉ (X → Y), is-equiv f × ((λ (q : Y → R) → f (ε (q ∘ f))) ＝ δ) characterization-of-selection-space-＝ : is-univalent 𝓤 → (A B : SelectionSpace) → (A ＝ B) ≃ (A ≅ B) characterization-of-selection-space-＝ ua = characterization-of-＝-with-axioms ua sns-data axioms axiomss
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
Here is an example where we need to refer to the inverse of the equivalence under consideration.
We take the opportunity to illustrate how the above boiler-plate code
can be avoided by defining sns-data on the fly, at the expense of
readability:
module contrived-example (𝓤 : Universe) where open sip contrived-＝ : is-univalent 𝓤 → (X Y : 𝓤 ̇ ) (φ : (X → X) → X) (γ : (Y → Y) → Y) → ((X , φ) ＝ (Y , γ)) ≃ (Σ f ꞉ (X → Y) , Σ i ꞉ is-equiv f , (λ (g : Y → Y) → f (φ (inverse f i ∘ g ∘ f))) ＝ γ) contrived-＝ ua X Y φ γ = characterization-of-＝ ua ((λ (X , φ) (Y , γ) (f , i) → (λ (g : Y → Y) → f (φ (inverse f i ∘ g ∘ f))) ＝ γ) , (λ (X , φ) → refl φ) , (λ φ γ → equivs-closed-under-∼ (id-is-equiv (φ ＝ γ)) (λ {(refl φ) → refl (refl φ)}))) (X , φ) (Y , γ)
Many of the above examples can be written in such a concise form.
We now characterize equality of functor algebras. In the following, we
don’t need to know that the functor preserves composition or to give
coherence data for the identification 𝓕-id.
module generalized-functor-algebra {𝓤 𝓥 : Universe} (F : 𝓤 ̇ → 𝓥 ̇ ) (𝓕 : {X Y : 𝓤 ̇ } → (X → Y) → F X → F Y) (𝓕-id : {X : 𝓤 ̇ } → 𝓕 (𝑖𝑑 X) ＝ 𝑖𝑑 (F X)) where open sip S : 𝓤 ̇ → 𝓤 ⊔ 𝓥 ̇ S X = F X → X sns-data : SNS S (𝓤 ⊔ 𝓥) sns-data = (ι , ρ , θ) where ι : (A B : Σ S) → ⟨ A ⟩ ≃ ⟨ B ⟩ → 𝓤 ⊔ 𝓥 ̇ ι (X , α) (Y , β) (f , _) = f ∘ α ＝ β ∘ 𝓕 f ρ : (A : Σ S) → ι A A (id-≃ ⟨ A ⟩) ρ (X , α) = α ＝⟨ ap (α ∘_) (𝓕-id ⁻¹) ⟩ α ∘ 𝓕 id ∎ θ : {X : 𝓤 ̇ } (α β : S X) → is-equiv (canonical-map ι ρ α β) θ {X} α β = γ where c : α ＝ β → α ＝ β ∘ 𝓕 id c = transport (α ＝_) (ρ (X , β)) i : is-equiv c i = transport-is-equiv (α ＝_) (ρ (X , β)) h : canonical-map ι ρ α β ∼ c h (refl _) = ρ (X , α) ＝⟨ refl-left ⁻¹ ⟩ refl α ∙ ρ (X , α) ∎ γ : is-equiv (canonical-map ι ρ α β) γ = equivs-closed-under-∼ i h characterization-of-functor-algebra-＝ : is-univalent 𝓤 → (X Y : 𝓤 ̇ ) (α : F X → X) (β : F Y → Y) → ((X , α) ＝ (Y , β)) ≃ (Σ f ꞉ (X → Y), is-equiv f × (f ∘ α ＝ β ∘ 𝓕 f)) characterization-of-functor-algebra-＝ ua X Y α β = characterization-of-＝ ua sns-data (X , α) (Y , β)
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity equivalence.
We now characterize equality of generalized preordered sets. This example is harder than the previous ones.
A type-valued preorder on a type X is a type-valued relation which
is reflexive and transitive. A type-valued, as opposed to a
subsingleton-valued preorder, could also be called an
∞-preorder.
type-valued-preorder-S : 𝓤 ̇ → 𝓤 ⊔ (𝓥 ⁺) ̇ type-valued-preorder-S {𝓤} {𝓥} X = Σ _≤_ ꞉ (X → X → 𝓥 ̇ ) , ((x : X) → x ≤ x) × ((x y z : X) → x ≤ y → y ≤ z → x ≤ z)
A category, also known as a
1-category, is a type-valued preorder subject to suitable axioms,
where the relation _≤_ is traditionally written hom, and where
identities are given by the reflexivity law, and composition is given
by the transitivity law.
We choose to use categorical notation and terminology for type-valued preorders.
module type-valued-preorder (𝓤 𝓥 : Universe) (ua : Univalence) where open sip fe : global-dfunext fe = univalence-gives-global-dfunext ua hfe : global-hfunext hfe = univalence-gives-global-hfunext ua S : 𝓤 ̇ → 𝓤 ⊔ (𝓥 ⁺) ̇ S = type-valued-preorder-S {𝓤} {𝓥} Type-valued-preorder : (𝓤 ⊔ 𝓥) ⁺ ̇ Type-valued-preorder = Σ S
But we will use the shorter notation Σ S in this submodule.
The type of objects of a type-valued preorder:
Ob : Σ S → 𝓤 ̇ Ob (X , homX , idX , compX ) = X
Its hom-types (or preorder):
hom : (𝓧 : Σ S) → Ob 𝓧 → Ob 𝓧 → 𝓥 ̇ hom (X , homX , idX , compX) = homX
Its identities (or reflexivities):
𝒾𝒹 : (𝓧 : Σ S) (x : Ob 𝓧) → hom 𝓧 x x 𝒾𝒹 (X , homX , idX , compX) = idX
Its composition law (or transitivity):
comp : (𝓧 : Σ S) (x y z : Ob 𝓧) → hom 𝓧 x y → hom 𝓧 y z → hom 𝓧 x z comp (X , homX , idX , compX) = compX
Notice that we have the so-called diagramatic order for composition.
The functoriality of a pair F , 𝓕 (where in category theory the
latter is also written F, by an abuse of
notation) says that
𝓕 preserves identities and composition:
functorial : (𝓧 𝓐 : Σ S) → (F : Ob 𝓧 → Ob 𝓐) → ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) → 𝓤 ⊔ 𝓥 ̇ functorial 𝓧 𝓐 F 𝓕' = pidentity × pcomposition where
In order to express the preservation of identities and composition in traditional form, we first define, locally, symbols for composition in applicative order, making the objects implicit:
_o_ : {x y z : Ob 𝓧} → hom 𝓧 y z → hom 𝓧 x y → hom 𝓧 x z g o f = comp 𝓧 _ _ _ f g _□_ : {a b c : Ob 𝓐} → hom 𝓐 b c → hom 𝓐 a b → hom 𝓐 a c g □ f = comp 𝓐 _ _ _ f g
And we also make implicit the object parameters of the action of the functor on arrows:
𝓕 : {x y : Ob 𝓧} → hom 𝓧 x y → hom 𝓐 (F x) (F y) 𝓕 f = 𝓕' _ _ f
Preservation of identities:
pidentity = (λ x → 𝓕 (𝒾𝒹 𝓧 x)) ＝ (λ x → 𝒾𝒹 𝓐 (F x))
Preservation of composition:
pcomposition = (λ x y z (f : hom 𝓧 x y) (g : hom 𝓧 y z) → 𝓕 (g o f)) ＝ (λ x y z (f : hom 𝓧 x y) (g : hom 𝓧 y z) → 𝓕 g □ 𝓕 f)
This time we will need two steps to characterize equality of type-valued preorders. The first one is as above, by considering a standard notion of structure:
sns-data : SNS S (𝓤 ⊔ (𝓥 ⁺)) sns-data = (ι , ρ , θ) where ι : (𝓧 𝓐 : Σ S) → ⟨ 𝓧 ⟩ ≃ ⟨ 𝓐 ⟩ → 𝓤 ⊔ (𝓥 ⁺) ̇ ι 𝓧 𝓐 (F , _) = Σ p ꞉ hom 𝓧 ＝ (λ x y → hom 𝓐 (F x) (F y)) , functorial 𝓧 𝓐 F (λ x y → transport (λ - → - x y) p) ρ : (𝓧 : Σ S) → ι 𝓧 𝓧 (id-≃ ⟨ 𝓧 ⟩) ρ 𝓧 = refl (hom 𝓧) , refl (𝒾𝒹 𝓧) , refl (comp 𝓧) θ : {X : 𝓤 ̇ } (s t : S X) → is-equiv (canonical-map ι ρ s t) θ {X} (homX , idX , compX) (homA , idA , compA) = g where φ = canonical-map ι ρ (homX , idX , compX) (homA , idA , compA) γ : codomain φ → domain φ γ (refl _ , refl _ , refl _) = refl _ η : γ ∘ φ ∼ id η (refl _) = refl _ ε : φ ∘ γ ∼ id ε (refl _ , refl _ , refl _) = refl _ g : is-equiv φ g = invertibles-are-equivs φ (γ , η , ε)
The above constructions are short thanks to computations-under-the-hood performed by Agda, and may require some effort to unravel.
The above automatically gives a characterization of equality of preorders. But this characterization uses another equality, of hom types. The second step translates this equality into an equivalence:
lemma : (𝓧 𝓐 : Σ S) (F : Ob 𝓧 → Ob 𝓐) → (Σ p ꞉ hom 𝓧 ＝ (λ x y → hom 𝓐 (F x) (F y)) , functorial 𝓧 𝓐 F (λ x y → transport (λ - → - x y) p)) ≃ (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y)) × functorial 𝓧 𝓐 F 𝓕) lemma 𝓧 𝓐 F = γ where e = (hom 𝓧 ＝ λ x y → hom 𝓐 (F x) (F y)) ≃⟨ i ⟩ (∀ x y → hom 𝓧 x y ＝ hom 𝓐 (F x) (F y)) ≃⟨ ii ⟩ (∀ x y → hom 𝓧 x y ≃ hom 𝓐 (F x) (F y)) ≃⟨ iii ⟩ (∀ x → Σ φ ꞉ (∀ y → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , ∀ y → is-equiv (φ y)) ≃⟨ iv ⟩ (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y))) ■ where i = hfunext₂-≃ hfe hfe (hom 𝓧 ) λ x y → hom 𝓐 (F x) (F y) ii = Π-cong fe fe (λ x → Π-cong fe fe (λ y → univalence-≃ (ua 𝓥) (hom 𝓧 x y) (hom 𝓐 (F x) (F y)))) iii = Π-cong fe fe (λ y → ΠΣ-distr-≃) iv = ΠΣ-distr-≃
Here Agda silently performs a laborious computation to accept the
definition of item v:
v : (p : hom 𝓧 ＝ λ x y → hom 𝓐 (F x) (F y)) → functorial 𝓧 𝓐 F (λ x y → transport (λ - → - x y) p) ≃ functorial 𝓧 𝓐 F (pr₁ (⌜ e ⌝ p)) v (refl _) = Id→Eq _ _ (refl _) γ = (Σ p ꞉ hom 𝓧 ＝ (λ x y → hom 𝓐 (F x) (F y)) , functorial 𝓧 𝓐 F (λ x y → transport (λ - → - x y) p)) ≃⟨ vi ⟩ (Σ p ꞉ hom 𝓧 ＝ (λ x y → hom 𝓐 (F x) (F y)) , functorial 𝓧 𝓐 F (pr₁ (⌜ e ⌝ p))) ≃⟨ vii ⟩ (Σ σ ꞉ (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y))) , functorial 𝓧 𝓐 F (pr₁ σ)) ≃⟨ viii ⟩ (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y)) × functorial 𝓧 𝓐 F 𝓕) ■ where vi = Σ-cong v vii = ≃-sym (Σ-change-of-variable _ ⌜ e ⌝ (⌜⌝-is-equiv e)) viii = Σ-assoc
Combining the two steps, we get the following characterization of equality of type-valued preorders in terms of equivalences:
characterization-of-type-valued-preorder-＝ : (𝓧 𝓐 : Σ S) → (𝓧 ＝ 𝓐) ≃ (Σ F ꞉ (Ob 𝓧 → Ob 𝓐) , is-equiv F × (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y)) × functorial 𝓧 𝓐 F 𝓕)) characterization-of-type-valued-preorder-＝ 𝓧 𝓐 = (𝓧 ＝ 𝓐) ≃⟨ i ⟩ (Σ F ꞉ (Ob 𝓧 → Ob 𝓐) , is-equiv F × (Σ p ꞉ hom 𝓧 ＝ (λ x y → hom 𝓐 (F x) (F y)) , functorial 𝓧 𝓐 F (λ x y → transport (λ - → - x y) p))) ≃⟨ ii ⟩ (Σ F ꞉ (Ob 𝓧 → Ob 𝓐) , is-equiv F × (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y)) × functorial 𝓧 𝓐 F 𝓕)) ■ where i = characterization-of-＝ (ua 𝓤) sns-data 𝓧 𝓐 ii = Σ-cong (λ F → Σ-cong (λ _ → lemma 𝓧 𝓐 F))
Exercise. The above equivalence is characterized by induction on identifications as the function that maps the reflexive identification to the identity functor.
Now we consider type-valued preorders subject to arbitrary axioms. The only reason we need to consider this explicitly is that again we need to combine two steps. The second step is the same, but the first step is modified to add axioms.
module type-valued-preorder-with-axioms (𝓤 𝓥 𝓦 : Universe) (ua : Univalence) (axioms : (X : 𝓤 ̇ ) → type-valued-preorder-S {𝓤} {𝓥} X → 𝓦 ̇ ) (axiomss : (X : 𝓤 ̇ ) (s : type-valued-preorder-S X) → is-subsingleton (axioms X s)) where open sip open sip-with-axioms open type-valued-preorder 𝓤 𝓥 ua S' : 𝓤 ̇ → 𝓤 ⊔ (𝓥 ⁺) ⊔ 𝓦 ̇ S' X = Σ s ꞉ S X , axioms X s sns-data' : SNS S' (𝓤 ⊔ (𝓥 ⁺)) sns-data' = add-axioms axioms axiomss sns-data
Recall that [_] is the map that forgets the axioms.
characterization-of-type-valued-preorder-＝-with-axioms : (𝓧' 𝓐' : Σ S') → (𝓧' ＝ 𝓐') ≃ (Σ F ꞉ (Ob [ 𝓧' ] → Ob [ 𝓐' ]) , is-equiv F × (Σ 𝓕 ꞉ ((x y : Ob [ 𝓧' ]) → hom [ 𝓧' ] x y → hom [ 𝓐' ] (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y)) × functorial [ 𝓧' ] [ 𝓐' ] F 𝓕)) characterization-of-type-valued-preorder-＝-with-axioms 𝓧' 𝓐' = (𝓧' ＝ 𝓐') ≃⟨ i ⟩ ([ 𝓧' ] ≃[ sns-data ] [ 𝓐' ]) ≃⟨ ii ⟩ _ ■ where i = characterization-of-＝-with-axioms (ua 𝓤) sns-data axioms axiomss 𝓧' 𝓐' ii = Σ-cong (λ F → Σ-cong (λ _ → lemma [ 𝓧' ] [ 𝓐' ] F))
We now characterize equality of categories. By choosing suitable axioms for type-valued preorders, we get categories:
module category (𝓤 𝓥 : Universe) (ua : Univalence) where open type-valued-preorder-with-axioms 𝓤 𝓥 (𝓤 ⊔ 𝓥) ua fe : global-dfunext fe = univalence-gives-global-dfunext ua S : 𝓤 ̇ → 𝓤 ⊔ (𝓥 ⁺) ̇ S = type-valued-preorder-S {𝓤} {𝓥}
The axioms say that
category-axioms : (X : 𝓤 ̇ ) → S X → 𝓤 ⊔ 𝓥 ̇ category-axioms X (homX , idX , compX) = hom-sets × identityl × identityr × associativity where _o_ : {x y z : X} → homX y z → homX x y → homX x z g o f = compX _ _ _ f g hom-sets = ∀ x y → is-set (homX x y) identityl = ∀ x y (f : homX x y) → f o (idX x) ＝ f identityr = ∀ x y (f : homX x y) → (idX y) o f ＝ f associativity = ∀ x y z t (f : homX x y) (g : homX y z) (h : homX z t) → (h o g) o f ＝ h o (g o f)
The first axiom is subsingleton valued because the property of being a set is a subsingleton type. The second and the third axioms are subsingleton valued in the presence of the first axiom, because equations between elements of sets are subsingletons, by definition of set. And because subsingletons are closed under products, the category axioms form a subsingleton type:
category-axioms-subsingleton : (X : 𝓤 ̇ ) (s : S X) → is-subsingleton (category-axioms X s) category-axioms-subsingleton X (homX , idX , compX) ca = γ ca where i : ∀ x y → is-set (homX x y) i = pr₁ ca γ : is-subsingleton (category-axioms X (homX , idX , compX)) γ = ×-is-subsingleton ss (×-is-subsingleton ls (×-is-subsingleton rs as)) where ss = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → being-set-is-subsingleton fe)) ls = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ f → i x y (compX x x y (idX x) f) f))) rs = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ f → i x y (compX x y y f (idX y)) f))) as = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → Π-is-subsingleton fe (λ z → Π-is-subsingleton fe (λ t → Π-is-subsingleton fe (λ f → Π-is-subsingleton fe (λ g → Π-is-subsingleton fe (λ h → i x t (compX x y t f (compX y z t g h)) (compX x z t (compX x y z f g) h))))))))
We are now ready to define the type of categories, as a subtype of that of type-valued preorders:
Cat : (𝓤 ⊔ 𝓥)⁺ ̇ Cat = Σ X ꞉ 𝓤 ̇ , Σ s ꞉ S X , category-axioms X s
We reuse of above names in a slightly different way, taking into account that now we have axioms, which we simply ignore:
Ob : Cat → 𝓤 ̇ Ob (X , (homX , idX , compX) , _) = X hom : (𝓧 : Cat) → Ob 𝓧 → Ob 𝓧 → 𝓥 ̇ hom (X , (homX , idX , compX) , _) = homX 𝒾𝒹 : (𝓧 : Cat) (x : Ob 𝓧) → hom 𝓧 x x 𝒾𝒹 (X , (homX , idX , compX) , _) = idX comp : (𝓧 : Cat) (x y z : Ob 𝓧) (f : hom 𝓧 x y) (g : hom 𝓧 y z) → hom 𝓧 x z comp (X , (homX , idX , compX) , _) = compX is-functorial : (𝓧 𝓐 : Cat) → (F : Ob 𝓧 → Ob 𝓐) → ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) → 𝓤 ⊔ 𝓥 ̇ is-functorial 𝓧 𝓐 F 𝓕' = pidentity × pcomposition where _o_ : {x y z : Ob 𝓧} → hom 𝓧 y z → hom 𝓧 x y → hom 𝓧 x z g o f = comp 𝓧 _ _ _ f g _□_ : {a b c : Ob 𝓐} → hom 𝓐 b c → hom 𝓐 a b → hom 𝓐 a c g □ f = comp 𝓐 _ _ _ f g 𝓕 : {x y : Ob 𝓧} → hom 𝓧 x y → hom 𝓐 (F x) (F y) 𝓕 f = 𝓕' _ _ f pidentity = (λ x → 𝓕 (𝒾𝒹 𝓧 x)) ＝ (λ x → 𝒾𝒹 𝓐 (F x)) pcomposition = (λ x y z (f : hom 𝓧 x y) (g : hom 𝓧 y z) → 𝓕 (g o f)) ＝ (λ x y z (f : hom 𝓧 x y) (g : hom 𝓧 y z) → 𝓕 g □ 𝓕 f)
Exercise. For type-valued preorders, functorial 𝓧 𝓐 F 𝓕 is not in
general a subsingleton, but for categories, is-functorial 𝓧 𝓐 F 𝓕
is always a subsingleton.
We now apply the module type-valued-preorder-with-axioms to
get the following characterization of identity of categories:
_⋍_ : Cat → Cat → 𝓤 ⊔ 𝓥 ̇ 𝓧 ⋍ 𝓐 = Σ F ꞉ (Ob 𝓧 → Ob 𝓐) , is-equiv F × (Σ 𝓕 ꞉ ((x y : Ob 𝓧) → hom 𝓧 x y → hom 𝓐 (F x) (F y)) , (∀ x y → is-equiv (𝓕 x y)) × is-functorial 𝓧 𝓐 F 𝓕) Id→EqCat : (𝓧 𝓐 : Cat) → 𝓧 ＝ 𝓐 → 𝓧 ⋍ 𝓐 Id→EqCat 𝓧 𝓧 (refl 𝓧) = 𝑖𝑑 (Ob 𝓧 ) , id-is-equiv (Ob 𝓧 ) , (λ x y → 𝑖𝑑 (hom 𝓧 x y)) , (λ x y → id-is-equiv (hom 𝓧 x y)) , refl (𝒾𝒹 𝓧) , refl (comp 𝓧) characterization-of-category-＝ : (𝓧 𝓐 : Cat) → (𝓧 ＝ 𝓐) ≃ (𝓧 ⋍ 𝓐) characterization-of-category-＝ = characterization-of-type-valued-preorder-＝-with-axioms category-axioms category-axioms-subsingleton Id→EqCat-is-equiv : (𝓧 𝓐 : Cat) → is-equiv (Id→EqCat 𝓧 𝓐) Id→EqCat-is-equiv 𝓧 𝓐 = equivs-closed-under-∼ (⌜⌝-is-equiv (characterization-of-category-＝ 𝓧 𝓐)) (γ 𝓧 𝓐) where γ : (𝓧 𝓐 : Cat) → Id→EqCat 𝓧 𝓐 ∼ ⌜ characterization-of-category-＝ 𝓧 𝓐 ⌝ γ 𝓧 𝓧 (refl 𝓧) = refl _
The HoTT book has a characterization of identity of categories as equivalence of categories in the traditional sense of category theory, assuming that the categories are univalent in a certain sense. We have chosen not to include the univalence requirement in our notion of category, although it may be argued that univalent category is the correct notion of category for univalent mathematics (because a univalent category may be equivalently defined as a category object in a 1-groupoid). In any case, the characterization of equality given here is not affected by the univalence requirement, or any subsingleton-valued property of categories.
The subsingleton truncation of a type X, if it exists, is the
universal solution of the problem of mapping X into a subsingleton
type. The purpose of this section is to make this precise.
The following is Voevosky’s approach to saying that a type is inhabited in such a way that the statement of inhabitation is a subsingleton, relying on function extensionality.
is-inhabited : 𝓤 ̇ → 𝓤 ⁺ ̇ is-inhabited {𝓤} X = (P : 𝓤 ̇ ) → is-subsingleton P → (X → P) → P
This says that if we have a function from X to a subsingleton P, then
P must have a point. So this fails when X=𝟘. Considering P=𝟘, we conclude
that ¬¬ X if X is inhabited, which says that X is non-empty.
For simplicity in the formulation of the theorems, we assume global function extensionality. A type can be pointed in many ways, but inhabited in at most one way:
inhabitation-is-subsingleton : global-dfunext → (X : 𝓤 ̇ ) → is-subsingleton (is-inhabited X) inhabitation-is-subsingleton fe X = Π-is-subsingleton fe (λ P → Π-is-subsingleton fe (λ (s : is-subsingleton P) → Π-is-subsingleton fe (λ (f : X → P) → s)))
The following is the introduction rule for inhabitation, which says that pointed types are inhabited:
inhabited-intro : {X : 𝓤 ̇ } → X → is-inhabited X inhabited-intro x = λ P s f → f x
And its recursion principle:
inhabited-recursion : {X P : 𝓤 ̇ } → is-subsingleton P → (X → P) → is-inhabited X → P inhabited-recursion s f φ = φ (codomain f) s f
And its computation rule:
inhabited-recursion-computation : {X P : 𝓤 ̇ } (i : is-subsingleton P) (f : X → P) (x : X) → inhabited-recursion i f (inhabited-intro x) ＝ f x inhabited-recursion-computation i f x = refl (f x)
So the point x inside inhabited x is available for use by any
function f into a subsingleton, via inhabited-recursion.
We can derive induction from recursion in this case, but its “computation rule” holds up to an identification, rather than judgmentally:
inhabited-induction : global-dfunext → {X : 𝓤 ̇ } {P : is-inhabited X → 𝓤 ̇ } (i : (s : is-inhabited X) → is-subsingleton (P s)) (f : (x : X) → P (inhabited-intro x)) → (s : is-inhabited X) → P s inhabited-induction fe {X} {P} i f s = φ' s where φ : X → P s φ x = transport P (inhabitation-is-subsingleton fe X (inhabited-intro x) s) (f x) φ' : is-inhabited X → P s φ' = inhabited-recursion (i s) φ inhabited-computation : (fe : global-dfunext) {X : 𝓤 ̇ } {P : is-inhabited X → 𝓤 ̇ } (i : (s : is-inhabited X) → is-subsingleton (P s)) (f : (x : X) → P (inhabited-intro x)) (x : X) → inhabited-induction fe i f (inhabited-intro x) ＝ f x inhabited-computation fe i f x = i (inhabited-intro x) (inhabited-induction fe i f (inhabited-intro x)) (f x)
The definition of inhabitation looks superficially like double negation.
However, although we don’t necessarily have that
¬¬ P → P, we do have that is-inhabited P → P if P is a subsingleton.
inhabited-subsingletons-are-pointed : (P : 𝓤 ̇ ) → is-subsingleton P → is-inhabited P → P inhabited-subsingletons-are-pointed P s = inhabited-recursion s (𝑖𝑑 P)
Exercise. Show that
is-inhabited X ⇔ ¬¬X if and only if excluded middle holds.
inhabited-functorial : global-dfunext → (X : 𝓤 ⁺ ̇ ) (Y : 𝓤 ̇ ) → (X → Y) → is-inhabited X → is-inhabited Y inhabited-functorial fe X Y f = inhabited-recursion (inhabitation-is-subsingleton fe Y) (inhabited-intro ∘ f)
This universe assignment for functoriality is fairly restrictive, but is the only possible one.
With this notion, we can define the image of a function as follows:
image' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → (𝓤 ⊔ 𝓥)⁺ ̇ image' f = Σ y ꞉ codomain f , is-inhabited (Σ x ꞉ domain f , f x ＝ y)
An attempt to define the image of f without the inhabitation
predicate would be to take it to be
Σ y ꞉ codomain f , Σ x ꞉ domain f , f x ＝ y.
But we already know that
this is equivalent to X. This is similar to what happens in set
theory: the graph of any function is in bijection with its domain.
We can define the restriction and corestriction of a function to its image as follows:
restriction' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → image' f → Y restriction' f (y , _) = y corestriction' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → X → image' f corestriction' f x = f x , inhabited-intro (x , refl (f x))
And we can define the notion of surjection as follows:
is-surjection' : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → (𝓤 ⊔ 𝓥)⁺ ̇ is-surjection' f = (y : codomain f) → is-inhabited (Σ x ꞉ domain f , f x ＝ y)
Exercise. The type
(y : codomain f) → Σ x ꞉ domain f , f x ＝ y is equivalent
to the type has-section f,
which is stronger than saying that f is a surjection.
There are two problems with this definition of inhabitation:
Inhabitation has values in the next universe.
We can eliminate into subsingletons of the same universe only.
In particular, it is not possible to show that the map X →
is-inhabited X is a surjection, or that X → Y gives is-inhabited X
→ is-inhabited Y for X and Y in arbitrary universes.
There are two proposed ways to solve this kind of problem:
Voevodsky works with certain resizing rules for subsingletons. At the time of writing, the (relative) consistency of the system with such rules is an open question.
The HoTT book works with certain higher inductive types (HIT’s), which are known to have models and hence to be (relatively) consistent. This is the same approach adopted by cubical type theory and cubical Agda.
A third possibility is to work with subsingleton truncations
axiomatically, which is compatible
with the above two proposals. We write this axiom as a record type
rather than as an iterated Σ type for simplicity, where we use the
HoTT-book notation ∥ X ∥ for the inhabitation of X,
called the propositional, or subsingleton, or truth-value, truncation of X:
record subsingleton-truncations-exist : 𝓤ω where field ∥_∥ : {𝓤 : Universe} → 𝓤 ̇ → 𝓤 ̇ ∥∥-is-subsingleton : {𝓤 : Universe} {X : 𝓤 ̇ } → is-subsingleton ∥ X ∥ ∣_∣ : {𝓤 : Universe} {X : 𝓤 ̇ } → X → ∥ X ∥ ∥∥-recursion : {𝓤 𝓥 : Universe} {X : 𝓤 ̇ } {P : 𝓥 ̇ } → is-subsingleton P → (X → P) → ∥ X ∥ → P infix 0 ∥_∥
This is the approach we adopt in our personal Agda development.
We now assume that subsingleton truncations exist in the next few
constructions, and we open the assumption to make the above fields
visible.
module basic-truncation-development (pt : subsingleton-truncations-exist) (hfe : global-hfunext) where open subsingleton-truncations-exist pt public hunapply : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {f g : Π A} → f ∼ g → f ＝ g hunapply = hfunext-gives-dfunext hfe ∥∥-recursion-computation : {X : 𝓤 ̇ } {P : 𝓥 ̇ } → (i : is-subsingleton P) → (f : X → P) → (x : X) → ∥∥-recursion i f ∣ x ∣ ＝ f x ∥∥-recursion-computation i f x = i (∥∥-recursion i f ∣ x ∣) (f x) ∥∥-induction : {X : 𝓤 ̇ } {P : ∥ X ∥ → 𝓥 ̇ } → ((s : ∥ X ∥) → is-subsingleton (P s)) → ((x : X) → P ∣ x ∣) → (s : ∥ X ∥) → P s ∥∥-induction {𝓤} {𝓥} {X} {P} i f s = φ' s where φ : X → P s φ x = transport P (∥∥-is-subsingleton ∣ x ∣ s) (f x) φ' : ∥ X ∥ → P s φ' = ∥∥-recursion (i s) φ ∥∥-computation : {X : 𝓤 ̇ } {P : ∥ X ∥ → 𝓥 ̇ } → (i : (s : ∥ X ∥) → is-subsingleton (P s)) → (f : (x : X) → P ∣ x ∣) → (x : X) → ∥∥-induction i f ∣ x ∣ ＝ f x ∥∥-computation i f x = i ∣ x ∣ (∥∥-induction i f ∣ x ∣) (f x) ∥∥-functor : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → ∥ X ∥ → ∥ Y ∥ ∥∥-functor f = ∥∥-recursion ∥∥-is-subsingleton (λ x → ∣ f x ∣)
The subsingleton truncation of a type and its inhabitation are logically equivalent propositions:
∥∥-agrees-with-inhabitation : (X : 𝓤 ̇ ) → ∥ X ∥ ⇔ is-inhabited X ∥∥-agrees-with-inhabitation X = a , b where a : ∥ X ∥ → is-inhabited X a = ∥∥-recursion (inhabitation-is-subsingleton hunapply X) inhabited-intro b : is-inhabited X → ∥ X ∥ b = inhabited-recursion ∥∥-is-subsingleton ∣_∣
Hence they differ only in size, and when size matters don’t get on the
way, we can use is-inhabited instead of ∥_∥ if we wish.
Disjunction and existence are defined as the truncation of + and Σ:
_∨_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ A ∨ B = ∥ A + B ∥ infixl 20 _∨_ ∃ : {X : 𝓤 ̇ } → (X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ ∃ A = ∥ Σ A ∥ -∃ : {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) (Y : X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ -∃ X Y = ∃ Y syntax -∃ A (λ x → b) = ∃ x ꞉ A , b infixr -1 -∃ ∨-is-subsingleton : {A : 𝓤 ̇ } {B : 𝓥 ̇ } → is-subsingleton (A ∨ B) ∨-is-subsingleton = ∥∥-is-subsingleton ∃-is-subsingleton : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } → is-subsingleton (∃ A) ∃-is-subsingleton = ∥∥-is-subsingleton
The author’s slides on univalent logic discuss further details about these notions of disjunction and existence.
The image of a function f : X → Y is the type of y : Y for which there exists x : X with f x ＝ y.
image : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ image f = Σ y ꞉ codomain f , ∃ x ꞉ domain f , f x ＝ y restriction : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → image f → Y restriction f (y , _) = y corestriction : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → X → image f corestriction f x = f x , ∣ (x , refl (f x)) ∣ is-surjection : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ is-surjection f = (y : codomain f) → ∃ x ꞉ domain f , f x ＝ y being-surjection-is-subsingleton : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-subsingleton (is-surjection f) being-surjection-is-subsingleton f = Π-is-subsingleton hunapply (λ y → ∃-is-subsingleton) corestriction-surjection : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-surjection (corestriction f) corestriction-surjection f (y , s) = ∥∥-functor g s where g : (Σ x ꞉ domain f , f x ＝ y) → Σ x ꞉ domain f , corestriction f x ＝ (y , s) g (x , p) = x , to-subtype-＝ (λ _ → ∃-is-subsingleton) p surjection-induction : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-surjection f → (P : Y → 𝓦 ̇ ) → ((y : Y) → is-subsingleton (P y)) → ((x : X) → P (f x)) → (y : Y) → P y surjection-induction f i P j α y = ∥∥-recursion (j y) φ (i y) where φ : fiber f y → P y φ (x , r) = transport P r (α x)
Exercise. A map is an equivalence if and only if it is both an embedding and a surjection. (To be solved shortly.)
This time we can prove that the map x ↦ ∣ x ∣ of X into ∥ X ∥ is
a surjection without the universe levels getting in our way:
∣∣-is-surjection : (X : 𝓤 ̇ ) → is-surjection (λ (x : X) → ∣ x ∣) ∣∣-is-surjection X s = γ where f : X → ∃ x ꞉ X , ∣ x ∣ ＝ s f x = ∣ (x , ∥∥-is-subsingleton ∣ x ∣ s) ∣ γ : ∃ x ꞉ X , ∣ x ∣ ＝ s γ = ∥∥-recursion ∥∥-is-subsingleton f s
Saying that this surjection X → ∥ X ∥ has a section for all X (we
can pick a point of every inhabited type) amounts to global
choice, which
contradicts univalence, and
also gives classical logic.
The subsingleton truncation of a type is also known as its
support, and a type X is
said to have split
support if there is a
choice function ∥ X ∥ → X, which is automatically a section of
the surjection X → ∥ X ∥.
Exercise. Show that a type has split support if and only it is logically equivalent to a subsingleton. In particular, the type of invertibility data has split support, as it is logically equivalent to the equivalence property.
Exercise (hard). If X and Y are types obtained by summing x- and
y-many copies of the type 𝟙, respectively, as in 𝟙 + 𝟙 + ... + 𝟙 , where x
and y are natural numbers, then ∥ X ＝ Y ∥ ≃ (x ＝ y) and the type
X ＝ X has x! elements.
Singletons are inhabited, of course:
singletons-are-inhabited : (X : 𝓤 ̇ ) → is-singleton X → ∥ X ∥ singletons-are-inhabited X s = ∣ center X s ∣
And inhabited subsingletons are singletons:
inhabited-subsingletons-are-singletons : (X : 𝓤 ̇ ) → ∥ X ∥ → is-subsingleton X → is-singleton X inhabited-subsingletons-are-singletons X t i = c , φ where c : X c = ∥∥-recursion i (𝑖𝑑 X) t φ : (x : X) → c ＝ x φ = i c
Hence a type is a singleton if and only if it is inhabited and a subsingleton:
singleton-iff-inhabited-subsingleton : (X : 𝓤 ̇ ) → is-singleton X ⇔ (∥ X ∥ × is-subsingleton X) singleton-iff-inhabited-subsingleton X = (λ (s : is-singleton X) → singletons-are-inhabited X s , singletons-are-subsingletons X s) , Σ-induction (inhabited-subsingletons-are-singletons X)
By considering the unique map X → 𝟙, this can be regarded as a
particular case of the fact that a map is an equivalence if and only
if it is both an embedding and a surjection:
equiv-iff-embedding-and-surjection : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → is-equiv f ⇔ (is-embedding f × is-surjection f) equiv-iff-embedding-and-surjection f = (a , b) where a : is-equiv f → is-embedding f × is-surjection f a e = (λ y → singletons-are-subsingletons (fiber f y) (e y)) , (λ y → singletons-are-inhabited (fiber f y) (e y)) b : is-embedding f × is-surjection f → is-equiv f b (e , s) y = inhabited-subsingletons-are-singletons (fiber f y) (s y) (e y) equiv-＝-embedding-and-surjection : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → propext (𝓤 ⊔ 𝓥) → is-equiv f ＝ (is-embedding f × is-surjection f) equiv-＝-embedding-and-surjection f pe = pe (being-equiv-is-subsingleton hunapply hunapply f) (×-is-subsingleton (being-embedding-is-subsingleton hunapply f) (being-surjection-is-subsingleton f)) (lr-implication (equiv-iff-embedding-and-surjection f)) (rl-implication (equiv-iff-embedding-and-surjection f))
We will see that global choice
(X : 𝓤 ̇ ) → ∥ X ∥ → X
is inconsistent with univalence, and also implies excluded
middle. However, for some types X, we can prove that ∥ X ∥ → X.
We characterize such types as those that have wconstant endomaps.
Because, as we have seen, we have a logical equivalence
∥ X ∥ ⇔ is-inhabited X,
it suffices to discuss
is-inhabited X → X,
which can be done in our spartan MLTT without any axioms for univalent mathematics (and hence also with axioms for univalent mathematics, including non-constructive ones such as excluded middle and choice).
For any type X, we have is-inhabited X → X iff X has a
designated wconstant
endomap. To prove this we first
show that the type of fixed points of a wconstant endomap is a
subsingleton.
We first define the type of fixed points:
fix : {X : 𝓤 ̇ } → (X → X) → 𝓤 ̇ fix f = Σ x ꞉ domain f , f x ＝ x
Of course any fixed point of f gives an element of X:
from-fix : {X : 𝓤 ̇ } (f : X → X) → fix f → X from-fix f = pr₁
Conversely, if f is wconstant then for any x : X we have that f
x is a fixed point of f, and hence from any element of X we get a
fixed point of f:
to-fix : {X : 𝓤 ̇ } (f : X → X) → wconstant f → X → fix f to-fix f κ x = f x , κ (f x) x
The following is trivial if the type X is a set. What may be
surprising is that it holds for arbitrary types, because in this case
the identity type f x ＝ x is in general not a subsingleton.
fix-is-subsingleton : {X : 𝓤 ̇ } (f : X → X) → wconstant f → is-subsingleton (fix f) fix-is-subsingleton {𝓤} {X} f κ = γ where a : (y x : X) → (f x ＝ x) ≃ (f y ＝ x) a y x = transport (_＝ x) (κ x y) , transport-is-equiv (_＝ x) (κ x y) b : (y : X) → fix f ≃ singleton-type' (f y) b y = Σ-cong (a y) c : X → is-singleton (fix f) c y = equiv-to-singleton (b y) (singleton-types'-are-singletons X (f y)) d : fix f → is-singleton (fix f) d = c ∘ from-fix f γ : is-subsingleton (fix f) γ = subsingleton-criterion d
Exercise. Formulate and prove the fact that the type fix f has the
universal property of the subsingleton truncation of X if f is
wconstant. Moreover, argue that the computation rule for recursion
holds definitionally in this case. This is an example of a situation
where the truncation of a type just is available in MLTT without
axioms or extensions.
We use fix-is-subsingleton to show that the type is-inhabited X →
X is logically equivalent to the type wconstant-endomap X, where
one direction uses function extensionality. We refer to a function
is-inhabited X → X as a choice function for X. So the claim is
that a type has a choice function if and only if it has a designated
wconstant endomap.
choice-function : 𝓤 ̇ → 𝓤 ⁺ ̇ choice-function X = is-inhabited X → X
With a constant endomap of X, we can exit the truncation
is-inhabited X in pure MLTT:
wconstant-endomap-gives-choice-function : {X : 𝓤 ̇ } → wconstant-endomap X → choice-function X wconstant-endomap-gives-choice-function {𝓤} {X} (f , κ) = from-fix f ∘ γ where γ : is-inhabited X → fix f γ = inhabited-recursion (fix-is-subsingleton f κ) (to-fix f κ)
For the converse we use function extensionality (to know that
the type is-inhabited X is a subsingleton in the construction of the wconstant
endomap):
choice-function-gives-wconstant-endomap : global-dfunext → {X : 𝓤 ̇ } → choice-function X → wconstant-endomap X choice-function-gives-wconstant-endomap fe {X} c = f , κ where f : X → X f = c ∘ inhabited-intro κ : wconstant f κ x y = ap c (inhabitation-is-subsingleton fe X (inhabited-intro x) (inhabited-intro y))
As an application, we show that if the type of roots of a function
f : ℕ → ℕ is inhabited, then it is pointed. In other words, with the
information that there is some root, we can find an explicit root.
module find-hidden-root where open basic-arithmetic-and-order public
Given a root, we find a minimal root (below it, of course) by bounded search, and this gives a constant endomap of the type of roots:
μρ : (f : ℕ → ℕ) → root f → root f μρ f r = minimal-root-is-root f (root-gives-minimal-root f r) μρ-root : (f : ℕ → ℕ) → root f → ℕ μρ-root f r = pr₁ (μρ f r) μρ-root-is-root : (f : ℕ → ℕ) (r : root f) → f (μρ-root f r) ＝ 0 μρ-root-is-root f r = pr₂ (μρ f r) μρ-root-minimal : (f : ℕ → ℕ) (m : ℕ) (p : f m ＝ 0) → (n : ℕ) → f n ＝ 0 → μρ-root f (m , p) ≤ n μρ-root-minimal f m p n q = not-<-gives-≥ (μρ-root f (m , p)) n γ where φ : ¬(f n ≠ 0) → ¬(n < μρ-root f (m , p)) φ = contrapositive (pr₂(pr₂ (root-gives-minimal-root f (m , p))) n) γ : ¬ (n < μρ-root f (m , p)) γ = φ (dni (f n ＝ 0) q)
The crucial property of the function μρ f is that it is wconstant:
μρ-wconstant : (f : ℕ → ℕ) → wconstant (μρ f) μρ-wconstant f (n , p) (n' , p') = r where m m' : ℕ m = μρ-root f (n , p) m' = μρ-root f (n' , p') l : m ≤ m' l = μρ-root-minimal f n p m' (μρ-root-is-root f (n' , p')) l' : m' ≤ m l' = μρ-root-minimal f n' p' m (μρ-root-is-root f (n , p)) q : m ＝ m' q = ≤-anti _ _ l l' r : μρ f (n , p) ＝ μρ f (n' , p') r = to-subtype-＝ (λ _ → ℕ-is-set (f _) 0) q
Using the wconstancy of μρ f, if a root of f exists, then we can
find one (which in fact will be the minimal one):
find-existing-root : (f : ℕ → ℕ) → is-inhabited (root f) → root f find-existing-root f = h ∘ g where γ : root f → fix (μρ f) γ = to-fix (μρ f) (μρ-wconstant f) g : is-inhabited (root f) → fix (μρ f) g = inhabited-recursion (fix-is-subsingleton (μρ f) (μρ-wconstant f)) γ h : fix (μρ f) → root f h = from-fix (μρ f)
In the following example, we first hide a root with
inhabited-intro and then find
the minimal root with search bounded by this hidden root:
module find-existing-root-example where f : ℕ → ℕ f 0 = 1 f 1 = 1 f 2 = 0 f 3 = 1 f 4 = 0 f 5 = 1 f 6 = 1 f 7 = 0 f (succ (succ (succ (succ (succ (succ (succ (succ x)))))))) = x
We hide the root 8 of f:
root-existence : is-inhabited (root f) root-existence = inhabited-intro (8 , refl 0) r : root f r = find-existing-root f root-existence x : ℕ x = pr₁ r x-is-root : f x ＝ 0 x-is-root = pr₂ r
We have that x evaluates to 2, which is clearly the minimal root
of f:
p : x ＝ 2 p = refl _
Thus, even though the type is-inhabited A is a subsingleton for any
type A, the function inhabited-intro : A → is-inhabited A doesn’t
erase information. We used the information contained in
root-existence as a bound for searching for the minimal root.
Notice that this construction is in pure (spartan) MLTT without
assumptions. Now we repeat part of the above using the existence of
small truncations and functional extensionality as assumptions,
replacing is-inhabited by ∥_∥:
module exit-∥∥ (pt : subsingleton-truncations-exist) (hfe : global-hfunext) where open basic-truncation-development pt hfe open find-hidden-root find-∥∥-existing-root : (f : ℕ → ℕ) → (∃ n ꞉ ℕ , f n ＝ 0) → Σ n ꞉ ℕ , f n ＝ 0 find-∥∥-existing-root f = k where γ : root f → fix (μρ f) γ = to-fix (μρ f) (μρ-wconstant f) g : ∥ root f ∥ → fix (μρ f) g = ∥∥-recursion (fix-is-subsingleton (μρ f) (μρ-wconstant f)) γ h : fix (μρ f) → root f h = from-fix (μρ f) k : ∥ root f ∥ → root f k = h ∘ g module find-∥∥-existing-root-example where f : ℕ → ℕ f 0 = 1 f 1 = 1 f 2 = 0 f 3 = 1 f 4 = 0 f 5 = 1 f 6 = 1 f 7 = 0 f (succ (succ (succ (succ (succ (succ (succ (succ x)))))))) = x root-∥∥-existence : ∥ root f ∥ root-∥∥-existence = ∣ 8 , refl 0 ∣ r : root f r = find-∥∥-existing-root f root-∥∥-existence x : ℕ x = pr₁ r x-is-root : f x ＝ 0 x-is-root = pr₂ r
This time, because the existence of propositional truncations is an
assumption for this submodule, we don’t have that x evaluates to
2, because the computation rule for truncation doesn’t hold
definitionally. But we do have that x is 2, applying the
computation rule manually.
NB : find-∥∥-existing-root f ＝ from-fix (μρ f) ∘ ∥∥-recursion (fix-is-subsingleton (μρ f) (μρ-wconstant f)) (to-fix (μρ f) (μρ-wconstant f)) NB = refl _ p : x ＝ 2 p = ap (pr₁ ∘ from-fix (μρ f)) (∥∥-recursion-computation (fix-is-subsingleton (μρ f) (μρ-wconstant f)) (to-fix (μρ f) (μρ-wconstant f)) (8 , refl _))
In cubical Agda, with the truncation defined as a higher inductive
type, x would compute to 2 automatically, like in our previous
example using Voevodsky’s truncation is-inhabited. This
concludes the example. We also have:
wconstant-endomap-gives-∥∥-choice-function : {X : 𝓤 ̇ } → wconstant-endomap X → (∥ X ∥ → X) wconstant-endomap-gives-∥∥-choice-function {𝓤} {X} (f , κ) = from-fix f ∘ γ where γ : ∥ X ∥ → fix f γ = ∥∥-recursion (fix-is-subsingleton f κ) (to-fix f κ) ∥∥-choice-function-gives-wconstant-endomap : {X : 𝓤 ̇ } → (∥ X ∥ → X) → wconstant-endomap X ∥∥-choice-function-gives-wconstant-endomap {𝓤} {X} c = f , κ where f : X → X f = c ∘ ∣_∣ κ : wconstant f κ x y = ap c (∥∥-is-subsingleton ∣ x ∣ ∣ y ∣)
There is another situation in which we can eliminate truncations that
is often useful in practice. The universal property of subsingleton
truncation says that we can get a function ∥ X ∥ → Y provided the
type Y is a subsingleton and we have a given function X →
Y. Because Y is a subsingleton, the given function is automatically
wconstant. Hence the following generalizes this to the situation in
which Y is a set:
∥∥-recursion-set : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ ) → is-set Y → (f : X → Y) → wconstant f → ∥ X ∥ → Y ∥∥-recursion-set {𝓤} {𝓥} X Y s f κ = f' where ψ : (y y' : Y) → (Σ x ꞉ X , f x ＝ y) → (Σ x' ꞉ X , f x' ＝ y') → y ＝ y' ψ y y' (x , r) (x' , r') = y ＝⟨ r ⁻¹ ⟩ f x ＝⟨ κ x x' ⟩ f x' ＝⟨ r' ⟩ y' ∎ φ : (y y' : Y) → (∃ x ꞉ X , f x ＝ y) → (∃ x' ꞉ X , f x' ＝ y') → y ＝ y' φ y y' u u' = ∥∥-recursion (s y y') (λ - → ∥∥-recursion (s y y') (ψ y y' -) u') u P : 𝓤 ⊔ 𝓥 ̇ P = image f i : is-subsingleton P i (y , u) (y' , u') = to-subtype-＝ (λ _ → ∃-is-subsingleton) (φ y y' u u') g : ∥ X ∥ → P g = ∥∥-recursion i (corestriction f) h : P → Y h = restriction f f' : ∥ X ∥ → Y f' = h ∘ g
If we try to do this with Voevodsky’s truncation is-inhabited, we
stumble into an insurmountable problem of size.
This section has that on rings as a
pre-requisite. We now apply the notion of subsingleton truncation to
give the promised examples of Noetherian rngs and commutative
Noetherian local rings. Subsingleton truncation is needed to have the
existential quantifier ∃ available, in order to be able to define
the notion of Noetherian ring.
module noetherian-local-ring (pt : subsingleton-truncations-exist) {𝓤 : Universe} (ua : Univalence) where open ring {𝓤} ua open basic-truncation-development pt hfe open ℕ-order
We first consider left Noetherian rngs and then Noetherian local rings.
For this we need the notion of left ideal of a rng 𝓡, which is an
element of the powerset 𝓟 ⟨ 𝓡 ⟩ of the underlying set ⟨ 𝓡 ⟩ of 𝓡:
is-left-ideal : (𝓡 : Rng) → 𝓟 ⟨ 𝓡 ⟩ → 𝓤 ̇ is-left-ideal (R , (_+_ , _·_) , (i , ii , iii , (O , _) , _)) I = (O ∈ I) × ((x y : R) → x ∈ I → y ∈ I → (x + y) ∈ I) × ((x y : R) → y ∈ I → (x · y) ∈ I) is-left-noetherian : (𝓡 : Rng) → 𝓤 ⁺ ̇ is-left-noetherian 𝓡 = (I : ℕ → 𝓟 ⟨ 𝓡 ⟩) → ((n : ℕ) → is-left-ideal 𝓡 (I n)) → ((n : ℕ) → I n ⊆ I (succ n)) → ∃ m ꞉ ℕ , ((n : ℕ) → m ≤ n → I m ＝ I n) LNRng : 𝓤 ⁺ ̇ LNRng = Σ 𝓡 ꞉ Rng , is-left-noetherian 𝓡
In order to be able to characterize equality of left Noetherian rngs, we
again need to show that is-left-noetherian is property rather than data:
being-ln-is-subsingleton : (𝓡 : Rng) → is-subsingleton (is-left-noetherian 𝓡) being-ln-is-subsingleton 𝓡 = Π-is-subsingleton fe (λ I → Π-is-subsingleton fe (λ _ → Π-is-subsingleton fe (λ _ → ∃-is-subsingleton))) forget-LN : LNRng → Rng forget-LN (𝓡 , _) = 𝓡 forget-LN-is-embedding : is-embedding forget-LN forget-LN-is-embedding = pr₁-is-embedding being-ln-is-subsingleton
Isomorphism of left Noetherian rngs:
_≅[LNRng]_ : LNRng → LNRng → 𝓤 ̇ ((R , (_+_ , _·_) , _) , _) ≅[LNRng] ((R' , (_+'_ , _·'_) , _) , _) = Σ f ꞉ (R → R') , is-equiv f × ((λ x y → f (x + y)) ＝ (λ x y → f x +' f y)) × ((λ x y → f (x · y)) ＝ (λ x y → f x ·' f y)) NB : (𝓡 𝓡' : LNRng) → (𝓡 ≅[LNRng] 𝓡') ＝ (forget-LN 𝓡 ≅[Rng] forget-LN 𝓡') NB 𝓡 𝓡' = refl _
Again the identity type of Noetherian rngs is in bijection with the type of Noetherian rng isomorphisms:
characterization-of-LNRng-＝ : (𝓡 𝓡' : LNRng) → (𝓡 ＝ 𝓡') ≃ (𝓡 ≅[LNRng] 𝓡') characterization-of-LNRng-＝ 𝓡 𝓡' = (𝓡 ＝ 𝓡') ≃⟨ i ⟩ (forget-LN 𝓡 ＝ forget-LN 𝓡') ≃⟨ ii ⟩ (𝓡 ≅[LNRng] 𝓡') ■ where i = ≃-sym (embedding-criterion-converse forget-LN forget-LN-is-embedding 𝓡 𝓡') ii = characterization-of-rng-＝ (forget-LN 𝓡) (forget-LN 𝓡')
Hence properties of left Noetherian rngs are invariant under
isomorphism. More generally, we can transport along type-valued
functions of left Noetherian rngs, with values in an arbitrary
universe 𝓥, rather than just truth-valued ones:
isomorphic-LNRng-transport : (A : LNRng → 𝓥 ̇ ) (𝓡 𝓡' : LNRng) → 𝓡 ≅[LNRng] 𝓡' → A 𝓡 → A 𝓡' isomorphic-LNRng-transport A 𝓡 𝓡' i a = a' where p : 𝓡 ＝ 𝓡' p = ⌜ ≃-sym (characterization-of-LNRng-＝ 𝓡 𝓡') ⌝ i a' : A 𝓡' a' = transport A p a
In particular, any theorem about a left Noetherian rng automatically applies to any left Noetherian rng isomorphic to it.
One can similarly define right Noetherian rng and Noetherian rng, and then obtain the expected characterizations of their identity types (exercise).
We now consider Noetherian local rings.
is-commutative : Rng → 𝓤 ̇ is-commutative (R , (_+_ , _·_) , _) = (x y : R) → x · y ＝ y · x being-commutative-is-subsingleton : (𝓡 : Rng) → is-subsingleton (is-commutative 𝓡) being-commutative-is-subsingleton (R , (_+_ , _·_) , i , ii-vii) = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (λ y → i (x · y) (y · x)))
In the presence of commutativity, there is no difference between left ideal, right ideal and two-sided ideal, and so one speaks of simply ideals. A commutative rng is said to be local if it has a unique maximal proper ideal.
is-Noetherian-Local : Ring → 𝓤 ⁺ ̇ is-Noetherian-Local (R , (𝟏 , _+_ , _·_) , i-vii , viii) = is-commutative 𝓡 × is-noetherian × is-local where 𝓡 : Rng 𝓡 = (R , (_+_ , _·_) , i-vii) is-ideal = is-left-ideal 𝓡 is-noetherian = is-left-noetherian 𝓡 is-proper-ideal : 𝓟 R → 𝓤 ̇ is-proper-ideal I = is-ideal I × (∃ x ꞉ ⟨ 𝓡 ⟩ , x ∉ I) is-local = ∃! I ꞉ 𝓟 R , is-proper-ideal I × ((J : 𝓟 R) → is-proper-ideal J → J ⊆ I) being-NL-is-subsingleton : (𝓡 : Ring) → is-subsingleton (is-Noetherian-Local 𝓡) being-NL-is-subsingleton (R , (𝟏 , _+_ , _·_) , i-vii , viii) = ×-is-subsingleton (being-commutative-is-subsingleton 𝓡) (×-is-subsingleton (being-ln-is-subsingleton 𝓡) (∃!-is-subsingleton _ fe)) where 𝓡 : Rng 𝓡 = (R , (_+_ , _·_) , i-vii) NL-Ring : 𝓤 ⁺ ̇ NL-Ring = Σ 𝓡 ꞉ Ring , is-Noetherian-Local 𝓡 _≅[NL]_ : NL-Ring → NL-Ring → 𝓤 ̇ ((R , (𝟏 , _+_ , _·_) , _) , _) ≅[NL] ((R' , (𝟏' , _+'_ , _·'_) , _) , _) = Σ f ꞉ (R → R') , is-equiv f × (f 𝟏 ＝ 𝟏') × ((λ x y → f (x + y)) ＝ (λ x y → f x +' f y)) × ((λ x y → f (x · y)) ＝ (λ x y → f x ·' f y)) forget-NL : NL-Ring → Ring forget-NL (𝓡 , _) = 𝓡 forget-NL-is-embedding : is-embedding forget-NL forget-NL-is-embedding = pr₁-is-embedding being-NL-is-subsingleton NB' : (𝓡 𝓡' : NL-Ring) → (𝓡 ≅[NL] 𝓡') ＝ (forget-NL 𝓡 ≅[Ring] forget-NL 𝓡') NB' 𝓡 𝓡' = refl _ characterization-of-NL-ring-＝ : (𝓡 𝓡' : NL-Ring) → (𝓡 ＝ 𝓡') ≃ (𝓡 ≅[NL] 𝓡') characterization-of-NL-ring-＝ 𝓡 𝓡' = (𝓡 ＝ 𝓡') ≃⟨ i ⟩ (forget-NL 𝓡 ＝ forget-NL 𝓡') ≃⟨ ii ⟩ (𝓡 ≅[NL] 𝓡') ■ where i = ≃-sym (embedding-criterion-converse forget-NL forget-NL-is-embedding 𝓡 𝓡') ii = characterization-of-ring-＝ (forget-NL 𝓡) (forget-NL 𝓡') isomorphic-NL-Ring-transport : (A : NL-Ring → 𝓥 ̇ ) (𝓡 𝓡' : NL-Ring) → 𝓡 ≅[NL] 𝓡' → A 𝓡 → A 𝓡' isomorphic-NL-Ring-transport A 𝓡 𝓡' i a = a' where p : 𝓡 ＝ 𝓡' p = ⌜ ≃-sym (characterization-of-NL-ring-＝ 𝓡 𝓡') ⌝ i a' : A 𝓡' a' = transport A p a
We remark that alternative definitions of the above notions are adopted for the purposes of constructive algebra.
We discuss unique choice, univalent choice and global choice.
A simple form of unique choice just holds in our spartan MLTT.
The full form of unique choice is logically equivalent to function extensionality.
Univalent choice implies excluded middle and is not provable or disprovable, but is consistent with univalence.
Global choice contradicts univalence, because it is not possible to choose an element of every inhabited type in way that is invariant under automorphisms.
The principle of simple unique choice says that
if for every
x : Xthere is a uniquea : A xwithR x a,
then
there is a specified function
f : (x : X) → A xsuch thatR x (f x)for allx : X.
This just holds and is trivial, given by projection:
simple-unique-choice : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (R : (x : X) → A x → 𝓦 ̇ ) → ((x : X) → ∃! a ꞉ A x , R x a) → Σ f ꞉ Π A , ((x : X) → R x (f x)) simple-unique-choice X A R s = f , φ where f : (x : X) → A x f x = pr₁ (center (Σ a ꞉ A x , R x a) (s x)) φ : (x : X) → R x (f x) φ x = pr₂ (center (Σ a ꞉ A x , R x a) (s x))
Below we also consider a
variation of simple unique
choice that works with ∃ (truncated Σ) rather than ∃!.
A full form of unique choice is Voevodsky’s formulation
vvfunext of function extensionality,
which says that products of singletons are singletons. We show that this
is equivalent to our official formulation of unique choice:
Unique-Choice : (𝓤 𝓥 𝓦 : Universe) → (𝓤 ⊔ 𝓥 ⊔ 𝓦)⁺ ̇ Unique-Choice 𝓤 𝓥 𝓦 = (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (R : (x : X) → A x → 𝓦 ̇ ) → ((x : X) → ∃! a ꞉ A x , R x a) → ∃! f ꞉ Π A , ((x : X) → R x (f x))
This can be read as saying that every single-valued relation is the graph of a unique function.
vvfunext-gives-unique-choice : vvfunext 𝓤 (𝓥 ⊔ 𝓦) → Unique-Choice 𝓤 𝓥 𝓦 vvfunext-gives-unique-choice vv X A R s = c where a : ((x : X) → Σ a ꞉ A x , R x a) ≃ (Σ f ꞉ ((x : X) → A x), ((x : X) → R x (f x))) a = ΠΣ-distr-≃ b : is-singleton ((x : X) → Σ a ꞉ A x , R x a) b = vv s c : is-singleton (Σ f ꞉ ((x : X) → A x), ((x : X) → R x (f x))) c = equiv-to-singleton' a b unique-choice-gives-vvfunext : Unique-Choice 𝓤 𝓥 𝓥 → vvfunext 𝓤 𝓥 unique-choice-gives-vvfunext {𝓤} {𝓥} uc {X} {A} φ = γ where R : (x : X) → A x → 𝓥 ̇ R x a = A x s' : (x : X) → is-singleton (A x × A x) s' x = ×-is-singleton (φ x) (φ x) s : (x : X) → ∃! y ꞉ A x , R x y s = s' e : ∃! f ꞉ Π A , ((x : X) → R x (f x)) e = uc X A R s e' : is-singleton (Π A × Π A) e' = e ρ : Π A ◁ Π A × Π A ρ = pr₁ , (λ y → y , y) , refl γ : is-singleton (Π A) γ = retract-of-singleton ρ e'
Here is an alternative proof that derives hfunext instead:
unique-choice-gives-hfunext : Unique-Choice 𝓤 𝓥 𝓥 → hfunext 𝓤 𝓥 unique-choice-gives-hfunext {𝓤} {𝓥} uc = →hfunext γ where γ : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (f : Π A) → ∃! g ꞉ Π A , f ∼ g γ X A f = uc X A R e where R : (x : X) → A x → 𝓥 ̇ R x a = f x ＝ a e : (x : X) → ∃! a ꞉ A x , f x ＝ a e x = singleton-types'-are-singletons (A x) (f x)
The above is not quite the converse of the previous, as there is a
universe mismatch, but we do get a logical equivalence by taking 𝓦
to be 𝓥:
unique-choice⇔vvfunext : Unique-Choice 𝓤 𝓥 𝓥 ⇔ vvfunext 𝓤 𝓥 unique-choice⇔vvfunext = unique-choice-gives-vvfunext , vvfunext-gives-unique-choice
We now give a different derivation of unique choice from function
extensionality, in order to illustrate transport along the inverse of
happly. For simplicity, we assume global function extensionality in
the next few constructions.
module _ (hfe : global-hfunext) where private hunapply : {X : 𝓤 ̇ } {A : X → 𝓥 ̇ } {f g : Π A} → f ∼ g → f ＝ g hunapply = inverse (happly _ _) (hfe _ _) transport-hunapply : {X : 𝓤 ̇ } (A : X → 𝓥 ̇ ) (R : (x : X) → A x → 𝓦 ̇ ) (f g : Π A) (φ : (x : X) → R x (f x)) (h : f ∼ g) (x : X) → transport (λ - → (x : X) → R x (- x)) (hunapply h) φ x ＝ transport (R x) (h x) (φ x) transport-hunapply A R f g φ h x = transport (λ - → ∀ x → R x (- x)) (hunapply h) φ x ＝⟨ i ⟩ transport (R x) (happly f g (hunapply h) x) (φ x) ＝⟨ ii ⟩ transport (R x) (h x) (φ x) ∎ where a : {f g : Π A} {φ : ∀ x → R x (f x)} (p : f ＝ g) (x : domain A) → transport (λ - → ∀ x → R x (- x)) p φ x ＝ transport (R x) (happly f g p x) (φ x) a (refl _) x = refl _ b : happly f g (hunapply h) ＝ h b = inverses-are-sections (happly f g) (hfe f g) h i = a (hunapply h) x ii = ap (λ - → transport (R x) (- x) (φ x)) b unique-choice : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (R : (x : X) → A x → 𝓦 ̇ ) → ((x : X) → ∃! a ꞉ A x , R x a) → ∃! f ꞉ ((x : X) → A x), ((x : X) → R x (f x)) unique-choice X A R s = C , Φ where f₀ : (x : X) → A x f₀ x = pr₁ (center (Σ a ꞉ A x , R x a) (s x)) φ₀ : (x : X) → R x (f₀ x) φ₀ x = pr₂ (center (Σ a ꞉ A x , R x a) (s x)) C : Σ f ꞉ ((x : X) → A x), ((x : X) → R x (f x)) C = f₀ , φ₀ c : (x : X) → (τ : Σ a ꞉ A x , R x a) → f₀ x , φ₀ x ＝ τ c x = centrality (Σ a ꞉ A x , R x a) (s x) c₁ : (x : X) (a : A x) (r : R x a) → f₀ x ＝ a c₁ x a r = ap pr₁ (c x (a , r)) c₂ : (x : X) (a : A x) (r : R x a) → transport (λ - → R x (pr₁ -)) (c x (a , r)) (φ₀ x) ＝ r c₂ x a r = apd pr₂ (c x (a , r)) Φ : (σ : Σ f ꞉ ((x : X) → A x), ((x : X) → R x (f x))) → C ＝ σ Φ (f , φ) = to-Σ-＝ (p , hunapply q) where p : f₀ ＝ f p = hunapply (λ x → c₁ x (f x) (φ x)) q : transport (λ - → (x : X) → R x (- x)) p φ₀ ∼ φ q x = transport (λ - → (x : X) → R x (- x)) p φ₀ x ＝⟨ i ⟩ transport (R x) (ap pr₁ (c x (f x , φ x))) (φ₀ x) ＝⟨ ii ⟩ transport (λ σ → R x (pr₁ σ)) (c x (f x , φ x)) (φ₀ x) ＝⟨ iii ⟩ φ x ∎ where i = transport-hunapply A R f₀ f φ₀ (λ x → c₁ x (f x) (φ x)) x ii = (transport-ap (R x) pr₁ (c x (f x , φ x)) (φ₀ x))⁻¹ iii = c₂ x (f x) (φ x)
Simple unique choice can be
reformulated as follows using ∃ rather than ∃!. The statement
is-subsingleton (Σ a ꞉ A x , R x a)
can be read as
there is at most one
a : A xwithR x a.
So the hypothesis of the following is that there is at most one such
a and at least one such a, which amounts to saying that there is a
unique such a, and hence simple-unique-choice' amounts to the same
thing as simple-unique-choice. However, simple-unique-choice can
be formulated and proved in our spartan MLTT, whereas
simple-unique-choice' requires the assumption of the existence of
subsingleton truncations so that ∃ is available for its formulation.
module choice (pt : subsingleton-truncations-exist) (hfe : global-hfunext) where open basic-truncation-development pt hfe public simple-unique-choice' : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) (R : (x : X) → A x → 𝓦 ̇ ) → ((x : X) → is-subsingleton (Σ a ꞉ A x , R x a)) → ((x : X) → ∃ a ꞉ A x , R x a) → Σ f ꞉ Π A , ((x : X) → R x (f x)) simple-unique-choice' X A R u φ = simple-unique-choice X A R s where s : (x : X) → ∃! a ꞉ A x , R x a s x = inhabited-subsingletons-are-singletons (Σ a ꞉ A x , R x a) (φ x) (u x)
In the next few subsections we continue working within the submodule
choice, in order to have the existence of propositional truncations
available, so that we can use the existential quantifier ∃.
The axiom of choice in univalent mathematics says that
if for every
x : Xthere existsa : A xwithR x a,
where R is some given relation,
then there exists a choice function
f : (x : X) → A xwithR x (f x)for allx : X,
provided
X is a set,A is a family of sets,R is subsingleton valued.This is not provable or disprovable in spartan univalent type theory,
but, with this proviso, it does hold in Voevodsky’s simplicial
model of our univalent type theory,
and hence is consistent. It is important that we have the condition
that A is a set-indexed family of sets and that the relation R is
subsingleton valued. For arbitrary higher groupoids, it is not in
general possible to perform the choice functorially.
AC : ∀ 𝓣 (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) → is-set X → ((x : X) → is-set (A x)) → 𝓣 ⁺ ⊔ 𝓤 ⊔ 𝓥 ̇ AC 𝓣 X A i j = (R : (x : X) → A x → 𝓣 ̇ ) → ((x : X) (a : A x) → is-subsingleton (R x a)) → ((x : X) → ∃ a ꞉ A x , R x a) → ∃ f ꞉ Π A , ((x : X) → R x (f x))
We define the axiom of choice in the universe 𝓤 to be the above with
𝓣 = 𝓤, for all possible X and A (and R).
Choice : ∀ 𝓤 → 𝓤 ⁺ ̇ Choice 𝓤 = (X : 𝓤 ̇ ) (A : X → 𝓤 ̇ ) (i : is-set X) (j : (x : X) → is-set (A x)) → AC 𝓤 X A i j
The above is equivalent to another familiar formulation of choice,
namely that a set-indexed product of non-empty sets is non-empty,
where in a constructive setting we strengthen non-empty to
inhabited (but this strengthening is immaterial, because choice
implies excluded middle, and excluded middle implies that
non-emptiness and inhabitation are the same notion).
IAC : (X : 𝓤 ̇ ) (Y : X → 𝓥 ̇ ) → is-set X → ((x : X) → is-set (Y x)) → 𝓤 ⊔ 𝓥 ̇ IAC X Y i j = ((x : X) → ∥ Y x ∥) → ∥ Π Y ∥ IChoice : ∀ 𝓤 → 𝓤 ⁺ ̇ IChoice 𝓤 = (X : 𝓤 ̇ ) (Y : X → 𝓤 ̇ ) (i : is-set X) (j : (x : X) → is-set (Y x)) → IAC X Y i j
These two forms of choice are logically equivalent (and hence equivalent, as both are subsingletons assuming function extensionality):
Choice-gives-IChoice : Choice 𝓤 → IChoice 𝓤 Choice-gives-IChoice {𝓤} ac X Y i j φ = γ where R : (x : X) → Y x → 𝓤 ̇ R x y = x ＝ x -- Any singleton type in 𝓤 will do. k : (x : X) (y : Y x) → is-subsingleton (R x y) k x y = i x x h : (x : X) → Y x → Σ y ꞉ Y x , R x y h x y = (y , refl x) g : (x : X) → ∃ y ꞉ Y x , R x y g x = ∥∥-functor (h x) (φ x) c : ∃ f ꞉ Π Y , ((x : X) → R x (f x)) c = ac X Y i j R k g γ : ∥ Π Y ∥ γ = ∥∥-functor pr₁ c IChoice-gives-Choice : IChoice 𝓤 → Choice 𝓤 IChoice-gives-Choice {𝓤} iac X A i j R k ψ = γ where Y : X → 𝓤 ̇ Y x = Σ a ꞉ A x , R x a l : (x : X) → is-set (Y x) l x = subsets-of-sets-are-sets (A x) (R x) (j x) (k x) a : ∥ Π Y ∥ a = iac X Y i l ψ h : Π Y → Σ f ꞉ Π A , ((x : X) → R x (f x)) h g = (λ x → pr₁ (g x)) , (λ x → pr₂ (g x)) γ : ∃ f ꞉ Π A , ((x : X) → R x (f x)) γ = ∥∥-functor h a
We will see that exiting trunctions in the sense of
(A : X → 𝓥 ̇ ) (x : X) → ∥ A x ∥ → A x
amounts to global choice and is inconsistent with univalence. However, exiting truncations in an unspecified way, in the sense of
(A : X → 𝓥 ̇ ) → ∥ (x : X) → ∥ A x ∥ → A x ∥
is consistent and equivalent to the above two versions of univalent choice, as we show now.
TAC : (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) → is-set X → ((x : X) → is-set (A x)) → 𝓤 ⊔ 𝓥 ̇ TAC X A i j = ∥((x : X) → ∥ A x ∥ → A x)∥ TChoice : ∀ 𝓤 → 𝓤 ⁺ ̇ TChoice 𝓤 = (X : 𝓤 ̇ ) (A : X → 𝓤 ̇ ) (i : is-set X) (j : (x : X) → is-set (A x)) → TAC X A i j
Notice that we use the hypothesis twice to get the conclusion in the following:
IChoice-gives-TChoice : IChoice 𝓤 → TChoice 𝓤 IChoice-gives-TChoice {𝓤} iac X A i j = γ where B : (x : X) → ∥ A x ∥ → 𝓤 ̇ B x s = A x k : (x : X) (s : ∥ A x ∥) → is-set (B x s) k x s = j x l : (x : X) → is-set ∥ A x ∥ l x = subsingletons-are-sets ∥ A x ∥ ∥∥-is-subsingleton m : (x : X) → is-set (∥ A x ∥ → A x) m x = Π-is-set hfe (λ s → j x) φ : (x : X) → ∥ (∥ A x ∥ → A x) ∥ φ x = iac ∥ A x ∥ (B x) (l x) (k x) (𝑖𝑑 ∥ A x ∥) γ : ∥((x : X) → ∥ A x ∥ → A x)∥ γ = iac X (λ x → ∥ A x ∥ → A x) i m φ TChoice-gives-IChoice : TChoice 𝓤 → IChoice 𝓤 TChoice-gives-IChoice tac X A i j = γ where γ : ((x : X) → ∥ A x ∥) → ∥ Π A ∥ γ g = ∥∥-functor φ (tac X A i j) where φ : ((x : X) → ∥ A x ∥ → A x) → Π A φ f x = f x (g x)
Exercise. A fourth formulation of the axiom of choice is that every surjection of sets has an unspecified section.
We apply the third formulation of univalent choice to show that choice implies excluded middle. We begin with the following lemma.
decidable-equality-criterion : {X : 𝓤 ̇ } (α : 𝟚 → X) → ((x : X) → (∃ n ꞉ 𝟚 , α n ＝ x) → (Σ n ꞉ 𝟚 , α n ＝ x)) → decidable (α ₀ ＝ α ₁) decidable-equality-criterion α c = γ d where r : 𝟚 → image α r = corestriction α σ : (y : image α) → Σ n ꞉ 𝟚 , r n ＝ y σ (x , t) = f u where u : Σ n ꞉ 𝟚 , α n ＝ x u = c x t f : (Σ n ꞉ 𝟚 , α n ＝ x) → Σ n ꞉ 𝟚 , r n ＝ (x , t) f (n , p) = n , to-subtype-＝ (λ _ → ∃-is-subsingleton) p s : image α → 𝟚 s y = pr₁ (σ y) η : (y : image α) → r (s y) ＝ y η y = pr₂ (σ y) l : left-cancellable s l = sections-are-lc s (r , η) αr : {m n : 𝟚} → α m ＝ α n → r m ＝ r n αr p = to-subtype-＝ (λ _ → ∃-is-subsingleton) p rα : {m n : 𝟚} → r m ＝ r n → α m ＝ α n rα = ap pr₁ αs : {m n : 𝟚} → α m ＝ α n → s (r m) ＝ s (r n) αs p = ap s (αr p) sα : {m n : 𝟚} → s (r m) ＝ s (r n) → α m ＝ α n sα p = rα (l p) γ : decidable (s (r ₀) ＝ s (r ₁)) → decidable(α ₀ ＝ α ₁) γ (inl p) = inl (sα p) γ (inr u) = inr (contrapositive αs u) d : decidable (s (r ₀) ＝ s (r ₁)) d = 𝟚-has-decidable-equality (s (r ₀)) (s (r ₁))
The first consequence of this lemma is that choice implies that every set has decidable equality.
choice-gives-decidable-equality : TChoice 𝓤 → (X : 𝓤 ̇ ) → is-set X → has-decidable-equality X choice-gives-decidable-equality {𝓤} tac X i x₀ x₁ = γ where α : 𝟚 → X α ₀ = x₀ α ₁ = x₁ A : X → 𝓤 ̇ A x = Σ n ꞉ 𝟚 , α n ＝ x l : is-subsingleton (decidable (x₀ ＝ x₁)) l = +-is-subsingleton' hunapply (i (α ₀) (α ₁)) δ : ∥((x : X) → ∥ A x ∥ → A x)∥ → decidable(x₀ ＝ x₁) δ = ∥∥-recursion l (decidable-equality-criterion α) j : (x : X) → is-set (A x) j x = subsets-of-sets-are-sets 𝟚 (λ n → α n ＝ x) 𝟚-is-set (λ n → i (α n) x) h : ∥((x : X) → ∥ A x ∥ → A x)∥ h = tac X A i j γ : decidable (x₀ ＝ x₁) γ = δ h
Applying the above to the object Ω 𝓤 of truth-values in the universe
𝓤, we get excluded middle:
choice-gives-EM : propext 𝓤 → TChoice (𝓤 ⁺) → EM 𝓤 choice-gives-EM {𝓤} pe tac = em where ⊤ : Ω 𝓤 ⊤ = (Lift 𝓤 𝟙 , equiv-to-subsingleton (Lift-≃ 𝟙) 𝟙-is-subsingleton) δ : (ω : Ω 𝓤) → decidable (⊤ ＝ ω) δ = choice-gives-decidable-equality tac (Ω 𝓤) (Ω-is-a-set hunapply pe) ⊤ em : (P : 𝓤 ̇ ) → is-subsingleton P → P + ¬ P em P i = γ (δ (P , i)) where γ : decidable (⊤ ＝ (P , i)) → P + ¬ P γ (inl r) = inl (Id→fun s (lift ⋆)) where s : Lift 𝓤 𝟙 ＝ P s = ap pr₁ r γ (inr n) = inr (contrapositive f n) where f : P → ⊤ ＝ P , i f p = Ω-ext hunapply pe (λ (_ : Lift 𝓤 𝟙) → p) (λ (_ : P) → lift ⋆)
For more information with Agda code, see this.
The following says that, for any given X, we can either choose a
point of X or tell that X is empty:
global-choice : (𝓤 : Universe) → 𝓤 ⁺ ̇ global-choice 𝓤 = (X : 𝓤 ̇ ) → X + is-empty X
And the following says that we can pick a point of every inhabited type:
global-∥∥-choice : (𝓤 : Universe) → 𝓤 ⁺ ̇ global-∥∥-choice 𝓤 = (X : 𝓤 ̇ ) → ∥ X ∥ → X
We first show that these two forms of global choice are logically equivalent, where one direction requires propositional extensionality (in addition to function extensionality, which is an assumption for this local module).
open exit-∥∥ pt hfe global-choice-gives-wconstant : global-choice 𝓤 → (X : 𝓤 ̇ ) → wconstant-endomap X global-choice-gives-wconstant g X = decidable-has-wconstant-endomap (g X) global-choice-gives-global-∥∥-choice : global-choice 𝓤 → global-∥∥-choice 𝓤 global-choice-gives-global-∥∥-choice {𝓤} c X = γ (c X) where γ : X + is-empty X → ∥ X ∥ → X γ (inl x) s = x γ (inr n) s = !𝟘 X (∥∥-recursion 𝟘-is-subsingleton n s) global-∥∥-choice-gives-all-types-are-sets : global-∥∥-choice 𝓤 → (X : 𝓤 ̇ ) → is-set X global-∥∥-choice-gives-all-types-are-sets {𝓤} c X = types-with-wconstant-＝-endomaps-are-sets X (λ x y → ∥∥-choice-function-gives-wconstant-endomap (c (x ＝ y))) global-∥∥-choice-gives-universe-is-set : global-∥∥-choice (𝓤 ⁺) → is-set (𝓤 ̇ ) global-∥∥-choice-gives-universe-is-set {𝓤} c = global-∥∥-choice-gives-all-types-are-sets c (𝓤 ̇ ) global-∥∥-choice-gives-choice : global-∥∥-choice 𝓤 → TChoice 𝓤 global-∥∥-choice-gives-choice {𝓤} c X A i j = ∣(λ x → c (A x))∣ global-∥∥-choice-gives-EM : propext 𝓤 → global-∥∥-choice (𝓤 ⁺) → EM 𝓤 global-∥∥-choice-gives-EM {𝓤} pe c = choice-gives-EM pe (global-∥∥-choice-gives-choice c) global-∥∥-choice-gives-global-choice : propext 𝓤 → global-∥∥-choice 𝓤 → global-∥∥-choice (𝓤 ⁺) → global-choice 𝓤 global-∥∥-choice-gives-global-choice {𝓤} pe c c⁺ X = γ where d : decidable ∥ X ∥ d = global-∥∥-choice-gives-EM pe c⁺ ∥ X ∥ ∥∥-is-subsingleton f : decidable ∥ X ∥ → X + is-empty X f (inl i) = inl (c X i) f (inr φ) = inr (contrapositive ∣_∣ φ) γ : X + is-empty X γ = f d
Two forms of globally global choice:
Global-Choice Global-∥∥-Choice : 𝓤ω Global-Choice = ∀ 𝓤 → global-choice 𝓤 Global-∥∥-Choice = ∀ 𝓤 → global-∥∥-choice 𝓤
Which are equivalent, where one direction uses propositional extensionality:
Global-Choice-gives-Global-∥∥-Choice : Global-Choice → Global-∥∥-Choice Global-Choice-gives-Global-∥∥-Choice c 𝓤 = global-choice-gives-global-∥∥-choice (c 𝓤) Global-∥∥-Choice-gives-Global-Choice : global-propext → Global-∥∥-Choice → Global-Choice Global-∥∥-Choice-gives-Global-Choice pe c 𝓤 = global-∥∥-choice-gives-global-choice pe (c 𝓤) (c (𝓤 ⁺))
And which are inconsistent with univalence:
global-∥∥-choice-inconsistent-with-univalence : Global-∥∥-Choice → Univalence → 𝟘 global-∥∥-choice-inconsistent-with-univalence g ua = γ (g 𝓤₁) (ua 𝓤₀) where open example-of-a-nonset γ : global-∥∥-choice 𝓤₁ → is-univalent 𝓤₀ → 𝟘 γ g ua = 𝓤₀-is-not-a-set ua (global-∥∥-choice-gives-universe-is-set g) global-choice-inconsistent-with-univalence : Global-Choice → Univalence → 𝟘 global-choice-inconsistent-with-univalence g = global-∥∥-choice-inconsistent-with-univalence (Global-Choice-gives-Global-∥∥-Choice g)
See also Theorem 3.2.2 and Corollary 3.2.7 of the HoTT book for a different argument that works with a single, arbitrary universe.
global-choice-gives-all-types-are-sets : global-choice 𝓤 → (X : 𝓤 ̇ ) → is-set X global-choice-gives-all-types-are-sets {𝓤} c X = hedberg (λ x y → c (x ＝ y))
Voevodsky considered resizing rules for a type theory for univalent foundations. These rules govern the syntax of the formal system, and hence are of a meta-mathematical nature.
Here we instead formulate, in our type theory without such rules, mathematical resizing principles. These principles are provable in the system with Voevodsky’s rules.
The consistency of the resizing rules is an open problem at the time of writing, but the resizing principles are consistent relative to ZFC with Grothendieck universes, because they follow from excluded middle, which is known to be validated by the simplicial-set model.
It is also an open problem whether the resizing principles discussed below have a computational interpretation.
We say that a type X has size 𝓥 if it is equivalent to a type in the
universe 𝓥:
_has-size_ : 𝓤 ̇ → (𝓥 : Universe) → 𝓥 ⁺ ⊔ 𝓤 ̇ X has-size 𝓥 = Σ Y ꞉ 𝓥 ̇ , X ≃ Y
The propositional resizing principle from a universe 𝓤 to a universe
𝓥 says that every subsingleton in 𝓤 has size 𝓥:
propositional-resizing : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥)⁺ ̇ propositional-resizing 𝓤 𝓥 = (P : 𝓤 ̇ ) → is-subsingleton P → P has-size 𝓥
We also consider global propositional resizing:
Propositional-resizing : 𝓤ω Propositional-resizing = {𝓤 𝓥 : Universe} → propositional-resizing 𝓤 𝓥
Resizing from a universe to a higher universe just holds, of course:
upper-resizing : ∀ {𝓤} 𝓥 (X : 𝓤 ̇ ) → X has-size (𝓤 ⊔ 𝓥) upper-resizing 𝓥 X = (Lift 𝓥 X , ≃-Lift X)
Moreover, the notion of size is upper closed:
has-size-is-upper : (X : 𝓤 ̇ ) → X has-size 𝓥 → X has-size (𝓥 ⊔ 𝓦) has-size-is-upper {𝓤} {𝓥} {𝓦} X (Y , e) = Z , c where Z : 𝓥 ⊔ 𝓦 ̇ Z = Lift 𝓦 Y d : Y ≃ Z d = ≃-Lift Y c : X ≃ Z c = e ● d upper-propositional-resizing : propositional-resizing 𝓤 (𝓤 ⊔ 𝓥) upper-propositional-resizing {𝓤} {𝓥} P i = upper-resizing 𝓥 P
We say that a type is small if it has a copy in the first universe:
is-small : 𝓤 ̇ → 𝓤 ⊔ 𝓤₁ ̇ is-small X = X has-size 𝓤₀
Then propositional resizing is equivalent to saying that all propositions of every universe are small:
all-propositions-are-small : ∀ 𝓤 → 𝓤 ⁺ ̇ all-propositions-are-small 𝓤 = (P : 𝓤 ̇ ) → is-prop P → is-small P all-propositions-are-small-means-PR₀ : all-propositions-are-small 𝓤 ＝ propositional-resizing 𝓤 𝓤₀ all-propositions-are-small-means-PR₀ = refl _ all-propositions-are-small-gives-PR : all-propositions-are-small 𝓤 → propositional-resizing 𝓤 𝓥 all-propositions-are-small-gives-PR {𝓤} {𝓥} a P i = γ where δ : P has-size 𝓤₀ δ = a P i γ : P has-size 𝓥 γ = has-size-is-upper P δ
A global version:
All-propositions-are-small : 𝓤ω All-propositions-are-small = ∀ 𝓤 → all-propositions-are-small 𝓤 PR-gives-All-propositions-are-small : Propositional-resizing → All-propositions-are-small PR-gives-All-propositions-are-small PR 𝓤 = PR All-propositions-are-small-gives-PR : All-propositions-are-small → Propositional-resizing All-propositions-are-small-gives-PR a {𝓤} {𝓥} = all-propositions-are-small-gives-PR (a 𝓤)
We use the following to work with propositional resizing more abstractly:
resize : propositional-resizing 𝓤 𝓥 → (P : 𝓤 ̇ ) (i : is-subsingleton P) → 𝓥 ̇ resize ρ P i = pr₁ (ρ P i) resize-is-subsingleton : (ρ : propositional-resizing 𝓤 𝓥) (P : 𝓤 ̇ ) (i : is-subsingleton P) → is-subsingleton (resize ρ P i) resize-is-subsingleton ρ P i = equiv-to-subsingleton (≃-sym (pr₂ (ρ P i))) i to-resize : (ρ : propositional-resizing 𝓤 𝓥) (P : 𝓤 ̇ ) (i : is-subsingleton P) → P → resize ρ P i to-resize ρ P i = ⌜ pr₂ (ρ P i) ⌝ from-resize : (ρ : propositional-resizing 𝓤 𝓥) (P : 𝓤 ̇ ) (i : is-subsingleton P) → resize ρ P i → P from-resize ρ P i = ⌜ ≃-sym(pr₂ (ρ P i)) ⌝
Propositional resizing is consistent because it is implied by excluded middle, which is consistent (with or without univalence):
EM-gives-all-propositions-are-small : EM 𝓤 → all-propositions-are-small 𝓤 EM-gives-all-propositions-are-small em P i = γ where Q : P + ¬ P → 𝓤₀ ̇ Q (inl _) = 𝟙 Q (inr _) = 𝟘 j : (d : P + ¬ P) → is-subsingleton (Q d) j (inl p) = 𝟙-is-subsingleton j (inr n) = 𝟘-is-subsingleton f : (d : P + ¬ P) → P → Q d f (inl _) _ = ⋆ f (inr n) p = !𝟘 𝟘 (n p) g : (d : P + ¬ P) → Q d → P g (inl p) _ = p g (inr _) q = !𝟘 P q e : P ≃ Q (em P i) e = logically-equivalent-subsingletons-are-equivalent P (Q (em P i)) i (j (em P i)) (f (em P i) , g (em P i)) γ : is-small P γ = Q (em P i) , e
Hence excluded middle implies propositional resizing:
EM-gives-PR : EM 𝓤 → propositional-resizing 𝓤 𝓥 EM-gives-PR {𝓤} {𝓥} em = all-propositions-are-small-gives-PR (EM-gives-all-propositions-are-small em)
To show that the propositional resizing principle is a subsingleton, we use univalence here.
has-size-is-subsingleton : Univalence → (X : 𝓤 ̇ ) (𝓥 : Universe) → is-subsingleton (X has-size 𝓥) has-size-is-subsingleton {𝓤} ua X 𝓥 = univalence→' (ua 𝓥) (ua (𝓤 ⊔ 𝓥)) X PR-is-subsingleton : Univalence → is-subsingleton (propositional-resizing 𝓤 𝓥) PR-is-subsingleton {𝓤} {𝓥} ua = Π-is-subsingleton (univalence-gives-global-dfunext ua) (λ P → Π-is-subsingleton (univalence-gives-global-dfunext ua) (λ i → has-size-is-subsingleton ua P 𝓥))
Exercise. It is possible to show that the propositional resizing principle is a subsingleton using propositional and functional extensionality instead of univalence.
We consider binary and unary notions of propositional impredicativity.
The binary notion of impredicativity, for two universes 𝓤 and 𝓥
says that the type of propositions in the universe 𝓤 has a copy in
the universe 𝓥.
Impredicativity : (𝓤 𝓥 : Universe) → (𝓤 ⊔ 𝓥 )⁺ ̇ Impredicativity 𝓤 𝓥 = Ω 𝓤 has-size 𝓥
The unary notion of impredicativity, for one 𝓤, says that the type
Ω 𝓤 of propositions, which lives in the universe 𝓤 ⁺, has a copy
in 𝓤 itself:
is-impredicative : (𝓤 : Universe) → 𝓤 ⁺ ̇ is-impredicative 𝓤 = Impredicativity 𝓤 𝓤
We can rephrase this in terms of a notion of relative smallness. We
say that a type X in a successor universe 𝓤 ⁺ is relatively
small if it has a copy in the universe 𝓤:
is-relatively-small : 𝓤 ⁺ ̇ → 𝓤 ⁺ ̇ is-relatively-small {𝓤} X = X has-size 𝓤 impredicativity-is-Ω-smallness : ∀ {𝓤} → is-impredicative 𝓤 ＝ is-relatively-small (Ω 𝓤) impredicativity-is-Ω-smallness {𝓤} = refl _
Propositional resizing gives impredicativity, in the presence of propositional and functional extensionality:
PR-gives-Impredicativity⁺ : global-propext → global-dfunext → propositional-resizing 𝓥 𝓤 → propositional-resizing 𝓤 𝓥 → Impredicativity 𝓤 (𝓥 ⁺) PR-gives-Impredicativity⁺ {𝓥} {𝓤} pe fe ρ σ = γ where φ : Ω 𝓥 → Ω 𝓤 φ (Q , j) = resize ρ Q j , resize-is-subsingleton ρ Q j ψ : Ω 𝓤 → Ω 𝓥 ψ (P , i) = resize σ P i , resize-is-subsingleton σ P i η : (p : Ω 𝓤) → φ (ψ p) ＝ p η (P , i) = Ω-ext fe pe a b where Q : 𝓥 ̇ Q = resize σ P i j : is-subsingleton Q j = resize-is-subsingleton σ P i a : resize ρ Q j → P a = from-resize σ P i ∘ from-resize ρ Q j b : P → resize ρ Q j b = to-resize ρ Q j ∘ to-resize σ P i ε : (q : Ω 𝓥) → ψ (φ q) ＝ q ε (Q , j) = Ω-ext fe pe a b where P : 𝓤 ̇ P = resize ρ Q j i : is-subsingleton P i = resize-is-subsingleton ρ Q j a : resize σ P i → Q a = from-resize ρ Q j ∘ from-resize σ P i b : Q → resize σ P i b = to-resize σ P i ∘ to-resize ρ Q j γ : (Ω 𝓤) has-size (𝓥 ⁺) γ = Ω 𝓥 , invertibility-gives-≃ ψ (φ , η , ε)
Propositional resizing doesn’t imply that the first universe 𝓤₀ is impredicative, but it does imply that all other, successor, universes 𝓤 ⁺ are.
PR-gives-impredicativity⁺ : global-propext → global-dfunext → propositional-resizing (𝓤 ⁺) 𝓤 → is-impredicative (𝓤 ⁺) PR-gives-impredicativity⁺ {𝓤} pe fe = PR-gives-Impredicativity⁺ pe fe (λ P i → upper-resizing (𝓤 ⁺) P)
What we get with propositional resizing is that all types of propositions of any universe 𝓤 are equivalent to Ω 𝓤₀, which lives in the second universe 𝓤₁:
PR-gives-impredicativity₁ : global-propext → global-dfunext → propositional-resizing 𝓤 𝓤₀ → Impredicativity 𝓤 𝓤₁ PR-gives-impredicativity₁ {𝓤} pe fe = PR-gives-Impredicativity⁺ pe fe (λ P i → upper-resizing 𝓤 P)
We can rephrase this as follows:
all-propositions-are-small-gives-impredicativity₁ : global-propext → global-dfunext → all-propositions-are-small 𝓤 → Ω 𝓤 has-size 𝓤₁ all-propositions-are-small-gives-impredicativity₁ = PR-gives-impredicativity₁
Exercise. Excluded middle gives the impredicativity of the first universe, and of all other universes.
We also have that moving Ω around universes moves subsingletons around
universes:
Impredicativity-gives-PR : propext 𝓤 → dfunext 𝓤 𝓤 → Impredicativity 𝓤 𝓥 → propositional-resizing 𝓤 𝓥 Impredicativity-gives-PR {𝓤} {𝓥} pe fe (O , e) P i = Q , ε where 𝟙' : 𝓤 ̇ 𝟙' = Lift 𝓤 𝟙 k : is-subsingleton 𝟙' k (lift ⋆) (lift ⋆) = refl (lift ⋆) down : Ω 𝓤 → O down = ⌜ e ⌝ O-is-set : is-set O O-is-set = equiv-to-set (≃-sym e) (Ω-is-a-set fe pe) Q : 𝓥 ̇ Q = down (𝟙' , k) ＝ down (P , i) j : is-subsingleton Q j = O-is-set (down (Lift 𝓤 𝟙 , k)) (down (P , i)) φ : Q → P φ q = Id→fun (ap _holds (equivs-are-lc down (⌜⌝-is-equiv e) q)) (lift ⋆) γ : P → Q γ p = ap down (to-subtype-＝ (λ _ → being-subsingleton-is-subsingleton fe) (pe k i (λ _ → p) (λ _ → lift ⋆))) ε : P ≃ Q ε = logically-equivalent-subsingletons-are-equivalent P Q i j (γ , φ)
Exercise. propext and funext and excluded middle together imply
that Ω 𝓤
has size 𝓤₀.
Using Voevodsky’s construction and propositional resizing, we get that function extensionality implies that subsingleton truncations exist:
PR-gives-existence-of-truncations : global-dfunext → Propositional-resizing → subsingleton-truncations-exist PR-gives-existence-of-truncations fe R = record { ∥_∥ = λ {𝓤} X → resize R (is-inhabited X) (inhabitation-is-subsingleton fe X) ; ∥∥-is-subsingleton = λ {𝓤} {X} → resize-is-subsingleton R (is-inhabited X) (inhabitation-is-subsingleton fe X) ; ∣_∣ = λ {𝓤} {X} x → to-resize R (is-inhabited X) (inhabitation-is-subsingleton fe X) (inhabited-intro x) ; ∥∥-recursion = λ {𝓤} {𝓥} {X} {P} i u s → from-resize R P i (inhabited-recursion (resize-is-subsingleton R P i) (to-resize R P i ∘ u) (from-resize R (is-inhabited X) (inhabitation-is-subsingleton fe X) s)) }
As a second, important, use of resizing, we revisit the powerset. First, given a set of subsets, that is, an element of the double powerset, we would like to consider its union. We investigate its existence in a submodule with assumptions.
module powerset-union-existence (pt : subsingleton-truncations-exist) (hfe : global-hfunext) where open basic-truncation-development pt hfe
Unions of families of subsets exist under the above assumption of
the existence of truncations, provided the index set I is in a universe
equal or below the universe of the type X we take the powerset of:
family-union : {X : 𝓤 ⊔ 𝓥 ̇ } {I : 𝓥 ̇ } → (I → 𝓟 X) → 𝓟 X family-union {𝓤} {𝓥} {X} {I} A = λ x → (∃ i ꞉ I , x ∈ A i) , ∃-is-subsingleton
Notice the increase of universe levels in the iteration of powersets:
𝓟𝓟 : 𝓤 ̇ → 𝓤 ⁺⁺ ̇ 𝓟𝓟 X = 𝓟 (𝓟 X)
The following doesn’t assert that unions of collections of subsets are available. It says what it means for them to be available:
existence-of-unions : (𝓤 : Universe) → 𝓤 ⁺⁺ ̇ existence-of-unions 𝓤 = (X : 𝓤 ̇ ) (𝓐 : 𝓟𝓟 X) → Σ B ꞉ 𝓟 X , ((x : X) → (x ∈ B) ⇔ (∃ A ꞉ 𝓟 X , (A ∈ 𝓐) × (x ∈ A)))
One may wonder whether there are different universe assignments for
the above definition, for instance whether we can instead assume 𝓐 :
(X → Ω 𝓥) → Ω 𝓦 for suitable universes 𝓥 and 𝓦, possibly
depending on 𝓤. That this is not the case can be checked by writing
the above definition in the following alternative form:
existence-of-unions₁ : (𝓤 : Universe) → _ ̇ existence-of-unions₁ 𝓤 = (X : 𝓤 ̇ ) (𝓐 : (X → Ω _) → Ω _) → Σ B ꞉ (X → Ω _) , ((x : X) → (x ∈ B) ⇔ (∃ A ꞉ (X → Ω _) , (A ∈ 𝓐) × (x ∈ A)))
The fact that Agda accepts the above definition without errors means
that there is a unique way to fill each _, which has to be the
following.
existence-of-unions₂ : (𝓤 : Universe) → 𝓤 ⁺⁺ ̇ existence-of-unions₂ 𝓤 = (X : 𝓤 ̇ ) (𝓐 : (X → Ω 𝓤) → Ω (𝓤 ⁺)) → Σ B ꞉ (X → Ω 𝓤) , ((x : X) → (x ∈ B) ⇔ (∃ A ꞉ (X → Ω 𝓤) , (A ∈ 𝓐) × (x ∈ A))) existence-of-unions-agreement : existence-of-unions 𝓤 ＝ existence-of-unions₂ 𝓤 existence-of-unions-agreement = refl _
To get the universe assigments explicitly, using Agda’s interactive
mode, we can write holes ? instead of _ and then ask Agda to fill
them uniquely by solving constraints, which is what we did to construct
existence-of-unions₂.
Exercise. Show that the existence of unions is a subsingleton type.
Without propositional resizing principles, it is not possible to establish the existence.
existence-of-unions-gives-PR : existence-of-unions 𝓤 → propositional-resizing (𝓤 ⁺) 𝓤 existence-of-unions-gives-PR {𝓤} α = γ where γ : (P : 𝓤 ⁺ ̇ ) → (i : is-subsingleton P) → P has-size 𝓤 γ P i = Q , e where 𝟙ᵤ : 𝓤 ̇ 𝟙ᵤ = Lift 𝓤 𝟙 ⋆ᵤ : 𝟙ᵤ ⋆ᵤ = lift ⋆ 𝟙ᵤ-is-subsingleton : is-subsingleton 𝟙ᵤ 𝟙ᵤ-is-subsingleton = equiv-to-subsingleton (Lift-≃ 𝟙) 𝟙-is-subsingleton 𝓐 : 𝓟𝓟 𝟙ᵤ 𝓐 = λ (A : 𝓟 𝟙ᵤ) → P , i B : 𝓟 𝟙ᵤ B = pr₁ (α 𝟙ᵤ 𝓐) φ : (x : 𝟙ᵤ) → (x ∈ B) ⇔ (∃ A ꞉ 𝓟 𝟙ᵤ , (A ∈ 𝓐) × (x ∈ A)) φ = pr₂ (α 𝟙ᵤ 𝓐) Q : 𝓤 ̇ Q = ⋆ᵤ ∈ B j : is-subsingleton Q j = ∈-is-subsingleton B ⋆ᵤ f : P → Q f p = b where a : Σ A ꞉ 𝓟 𝟙ᵤ , (A ∈ 𝓐) × (⋆ᵤ ∈ A) a = (λ (x : 𝟙ᵤ) → 𝟙ᵤ , 𝟙ᵤ-is-subsingleton) , p , ⋆ᵤ b : ⋆ᵤ ∈ B b = rl-implication (φ ⋆ᵤ) ∣ a ∣ g : Q → P g q = ∥∥-recursion i b a where a : ∃ A ꞉ 𝓟 𝟙ᵤ , (A ∈ 𝓐) × (⋆ᵤ ∈ A) a = lr-implication (φ ⋆ᵤ) q b : (Σ A ꞉ 𝓟 𝟙ᵤ , (A ∈ 𝓐) × (⋆ᵤ ∈ A)) → P b (A , m , _) = m e : P ≃ Q e = logically-equivalent-subsingletons-are-equivalent P Q i j (f , g)
The converse also holds, with an easier construction:
PR-gives-existence-of-unions : propositional-resizing (𝓤 ⁺) 𝓤 → existence-of-unions 𝓤 PR-gives-existence-of-unions {𝓤} ρ X 𝓐 = B , (λ x → lr x , rl x) where β : X → 𝓤 ⁺ ̇ β x = ∃ A ꞉ 𝓟 X , (A ∈ 𝓐) × (x ∈ A) i : (x : X) → is-subsingleton (β x) i x = ∃-is-subsingleton B : 𝓟 X B x = (resize ρ (β x) (i x) , resize-is-subsingleton ρ (β x) (i x)) lr : (x : X) → x ∈ B → ∃ A ꞉ 𝓟 X , (A ∈ 𝓐) × (x ∈ A) lr x = from-resize ρ (β x) (i x) rl : (x : X) → (∃ A ꞉ 𝓟 X , (A ∈ 𝓐) × (x ∈ A)) → x ∈ B rl x = to-resize ρ (β x) (i x)
We now close the above submodule and start another one with different assumptions:
module basic-powerset-development (hfe : global-hfunext) (ρ : Propositional-resizing) where pt : subsingleton-truncations-exist pt = PR-gives-existence-of-truncations (hfunext-gives-dfunext hfe) ρ open basic-truncation-development pt hfe open powerset-union-existence pt hfe ⋃ : {X : 𝓤 ̇ } → 𝓟𝓟 X → 𝓟 X ⋃ 𝓐 = pr₁ (PR-gives-existence-of-unions ρ _ 𝓐) ⋃-property : {X : 𝓤 ̇ } (𝓐 : 𝓟𝓟 X) → (x : X) → (x ∈ ⋃ 𝓐) ⇔ (∃ A ꞉ 𝓟 X , (A ∈ 𝓐) × (x ∈ A)) ⋃-property 𝓐 = pr₂ (PR-gives-existence-of-unions ρ _ 𝓐)
The construction of intersections is as that of unions using propositional resizing:
intersections-exist : (X : 𝓤 ̇ ) (𝓐 : 𝓟𝓟 X) → Σ B ꞉ 𝓟 X , ((x : X) → (x ∈ B) ⇔ ((A : 𝓟 X) → A ∈ 𝓐 → x ∈ A)) intersections-exist {𝓤} X 𝓐 = B , (λ x → lr x , rl x) where β : X → 𝓤 ⁺ ̇ β x = (A : 𝓟 X) → A ∈ 𝓐 → x ∈ A i : (x : X) → is-subsingleton (β x) i x = Π-is-subsingleton hunapply (λ A → Π-is-subsingleton hunapply (λ _ → ∈-is-subsingleton A x)) B : 𝓟 X B x = (resize ρ (β x) (i x) , resize-is-subsingleton ρ (β x) (i x)) lr : (x : X) → x ∈ B → (A : 𝓟 X) → A ∈ 𝓐 → x ∈ A lr x = from-resize ρ (β x) (i x) rl : (x : X) → ((A : 𝓟 X) → A ∈ 𝓐 → x ∈ A) → x ∈ B rl x = to-resize ρ (β x) (i x) ⋂ : {X : 𝓤 ̇ } → 𝓟𝓟 X → 𝓟 X ⋂ {𝓤} {X} 𝓐 = pr₁ (intersections-exist X 𝓐) ⋂-property : {X : 𝓤 ̇ } (𝓐 : 𝓟𝓟 X) → (x : X) → (x ∈ ⋂ 𝓐) ⇔ ((A : 𝓟 X) → A ∈ 𝓐 → x ∈ A) ⋂-property {𝓤} {X} 𝓐 = pr₂ (intersections-exist X 𝓐) ∅ full : {X : 𝓤 ̇ } → 𝓟 X ∅ = λ x → (Lift _ 𝟘 , equiv-to-subsingleton (Lift-≃ 𝟘) 𝟘-is-subsingleton) full = λ x → (Lift _ 𝟙 , equiv-to-subsingleton (Lift-≃ 𝟙) 𝟙-is-subsingleton) ∅-property : (X : 𝓤 ̇ ) → (x : X) → ¬(x ∈ ∅) ∅-property X x = lower full-property : (X : 𝓤 ̇ ) → (x : X) → x ∈ full full-property X x = lift ⋆ _∩_ _∪_ : {X : 𝓤 ̇ } → 𝓟 X → 𝓟 X → 𝓟 X (A ∪ B) = λ x → ((x ∈ A) ∨ (x ∈ B)) , ∨-is-subsingleton (A ∩ B) = λ x → ((x ∈ A) × (x ∈ B)) , ×-is-subsingleton (∈-is-subsingleton A x) (∈-is-subsingleton B x) ∪-property : {X : 𝓤 ̇ } (A B : 𝓟 X) → (x : X) → x ∈ (A ∪ B) ⇔ (x ∈ A) ∨ (x ∈ B) ∪-property {𝓤} {X} A B x = id , id ∩-property : {X : 𝓤 ̇ } (A B : 𝓟 X) → (x : X) → x ∈ (A ∩ B) ⇔ (x ∈ A) × (x ∈ B) ∩-property {𝓤} {X} A B x = id , id infix 20 _∩_ infix 20 _∪_
For example, with this we can define the type of topological spaces as
follows, where 𝓞 consists of designated sets, conventionally called
open and collectively referred to as the topology on X, which are
stipulated to be closed under finite intersections and arbitrary
unions. For finite intersections we consider the unary case full and
the binary case ∩ . Because the empty set is the union of the empty
collection (exercise), it is automatically included among the open
sets.
Top : (𝓤 : Universe) → 𝓤 ⁺⁺ ̇ Top 𝓤 = Σ X ꞉ 𝓤 ̇ , is-set X × (Σ 𝓞 ꞉ 𝓟𝓟 X , full ∈ 𝓞 × ((U V : 𝓟 X) → U ∈ 𝓞 → V ∈ 𝓞 → (U ∩ V) ∈ 𝓞) × ((𝓖 : 𝓟𝓟 X) → 𝓖 ⊆ 𝓞 → ⋃ 𝓖 ∈ 𝓞))
Notice that this jumps two universes. It is also possible, with
Ω-resizing, to construct the powerset in such a way that the powerset
of any type lives in the same universe as the type (exercise), and
hence so that the type of topological spaces in a base universe lives
in the next universe (exercise), rather than two universes above the
base universe.
Exercise. For a function X → Y, define its inverse image 𝓟 Y → 𝓟
X and its direct image 𝓟 X → 𝓟 Y. Define the notion of a
continuous map of topological spaces, namely a function of the
underlying sets whose inverse images of open sets are open. Show that
the identity function is continuous and that continuous maps are
closed under composition.
We now construct quotients using a technique proposed by Voevodsky, who assumed propositional resizing for that purpose, so that the quotient of a given type by a given equivalence relation would live in the same universe as the type. But the requirement that the quotient lives in the same universe is not needed to prove the universal property of the quotient.
We construct the quotient using propositional truncations, assuming functional and propositional extensionality, without assuming resizing.
A binary relation _≈_ on a type X : 𝓤 with values in a universe
𝓥 (which can of course be 𝓤) is called an equivalence relation
if it is subsingleton-valued, reflexive, symmetric and transitive.
All these notions
is-subsingleton-valued reflexive symmetric transitive is-equivalence-relation :
have the same type
{X : 𝓤 ̇ } → (X → X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇
and are defined by
is-subsingleton-valued _≈_ = ∀ x y → is-subsingleton (x ≈ y) reflexive _≈_ = ∀ x → x ≈ x symmetric _≈_ = ∀ x y → x ≈ y → y ≈ x transitive _≈_ = ∀ x y z → x ≈ y → y ≈ z → x ≈ z is-equivalence-relation _≈_ = is-subsingleton-valued _≈_ × reflexive _≈_ × symmetric _≈_ × transitive _≈_
We now work with a submodule with parameters to quotient a given type
X by a given equivalence relation _≈_. We assume not only the
existence of propositional truncations, as discussed above, but also
functional and propositional extensionality.
module quotient {𝓤 𝓥 : Universe} (pt : subsingleton-truncations-exist) (hfe : global-hfunext) (pe : propext 𝓥) (X : 𝓤 ̇ ) (_≈_ : X → X → 𝓥 ̇ ) (≈p : is-subsingleton-valued _≈_) (≈r : reflexive _≈_) (≈s : symmetric _≈_) (≈t : transitive _≈_) where open basic-truncation-development pt hfe
From the given relation
_≈_ : X → X → 𝓥 ̇
we define a function
X → (X → Ω 𝓥),
and we take the quotient X/≈ to be the image of this function. It is
for constructing the image that we need subsingleton
truncations. Functional and propositional extensionality are then used
to prove that the quotient is a set.
equiv-rel : X → (X → Ω 𝓥) equiv-rel x y = (x ≈ y) , ≈p x y X/≈ : 𝓥 ⁺ ⊔ 𝓤 ̇ X/≈ = image equiv-rel X/≈-is-set : is-set X/≈ X/≈-is-set = subsets-of-sets-are-sets (X → Ω 𝓥) _ (powersets-are-sets (dfunext-gives-hfunext hunapply) hunapply pe) (λ _ → ∃-is-subsingleton) η : X → X/≈ η = corestriction equiv-rel
We show that η is the universal solution to the problem of transforming
equivalence _≈_ into equality _＝_.
By construction, η is a surjection, of course:
η-surjection : is-surjection η η-surjection = corestriction-surjection equiv-rel
It is convenient to use the following induction principle for
reasoning about the image X/≈.
η-induction : (P : X/≈ → 𝓦 ̇ ) → ((x' : X/≈) → is-subsingleton (P x')) → ((x : X) → P (η x)) → (x' : X/≈) → P x' η-induction = surjection-induction η η-surjection
The first part of the universal property of η says that equivalent
points are mapped to identified points:
η-equiv-equal : {x y : X} → x ≈ y → η x ＝ η y η-equiv-equal {x} {y} e = to-subtype-＝ (λ _ → ∃-is-subsingleton) (hunapply (λ z → to-subtype-＝ (λ _ → being-subsingleton-is-subsingleton hunapply) (pe (≈p x z) (≈p y z) (≈t y x z (≈s x y e)) (≈t x y z e))))
To prove the required universal property, we also need the fact that
η reflects equality into equivalence:
η-equal-equiv : {x y : X} → η x ＝ η y → x ≈ y η-equal-equiv {x} {y} p = equiv-rel-reflect (ap pr₁ p) where equiv-rel-reflect : equiv-rel x ＝ equiv-rel y → x ≈ y equiv-rel-reflect q = b (≈r y) where a : (y ≈ y) ＝ (x ≈ y) a = ap (λ - → pr₁(- y)) (q ⁻¹) b : y ≈ y → x ≈ y b = Id→fun a
We are now ready to formulate and prove the required universal
property of the quotient. What is noteworthy here, regarding
universes, is that the universal property says that we can eliminate
into any set A of any universe 𝓦.
universal-property : (A : 𝓦 ̇ ) → is-set A → (f : X → A) → ({x x' : X} → x ≈ x' → f x ＝ f x') → ∃! f' ꞉ (X/≈ → A), f' ∘ η ＝ f universal-property {𝓦} A i f τ = e where G : X/≈ → 𝓥 ⁺ ⊔ 𝓤 ⊔ 𝓦 ̇ G x' = Σ a ꞉ A , ∃ x ꞉ X , (η x ＝ x') × (f x ＝ a) φ : (x' : X/≈) → is-subsingleton (G x') φ = η-induction _ γ induction-step where induction-step : (y : X) → is-subsingleton (G (η y)) induction-step x (a , d) (b , e) = to-subtype-＝ (λ _ → ∃-is-subsingleton) p where h : (Σ x' ꞉ X , (η x' ＝ η x) × (f x' ＝ a)) → (Σ y' ꞉ X , (η y' ＝ η x) × (f y' ＝ b)) → a ＝ b h (x' , r , s) (y' , t , u) = a ＝⟨ s ⁻¹ ⟩ f x' ＝⟨ τ (η-equal-equiv (r ∙ t ⁻¹)) ⟩ f y' ＝⟨ u ⟩ b ∎ p : a ＝ b p = ∥∥-recursion (i a b) (λ σ → ∥∥-recursion (i a b) (h σ) e) d γ : (x' : X/≈) → is-subsingleton (is-subsingleton (G x')) γ x' = being-subsingleton-is-subsingleton hunapply k : (x' : X/≈) → G x' k = η-induction _ φ induction-step where induction-step : (y : X) → G (η y) induction-step x = f x , ∣ x , refl (η x) , refl (f x) ∣ f' : X/≈ → A f' x' = pr₁ (k x') r : f' ∘ η ＝ f r = hunapply h where g : (y : X) → ∃ x ꞉ X , (η x ＝ η y) × (f x ＝ f' (η y)) g y = pr₂ (k (η y)) j : (y : X) → (Σ x ꞉ X , (η x ＝ η y) × (f x ＝ f' (η y))) → f'(η y) ＝ f y j y (x , p , q) = f' (η y) ＝⟨ q ⁻¹ ⟩ f x ＝⟨ τ (η-equal-equiv p) ⟩ f y ∎ h : (y : X) → f'(η y) ＝ f y h y = ∥∥-recursion (i (f' (η y)) (f y)) (j y) (g y) c : (σ : Σ f'' ꞉ (X/≈ → A), f'' ∘ η ＝ f) → (f' , r) ＝ σ c (f'' , s) = to-subtype-＝ (λ g → Π-is-set hfe (λ _ → i) (g ∘ η) f) t where w : ∀ x → f'(η x) ＝ f''(η x) w = happly (f' ∘ η) (f'' ∘ η) (r ∙ s ⁻¹) t : f' ＝ f'' t = hunapply (η-induction _ (λ x' → i (f' x') (f'' x')) w) e : ∃! f' ꞉ (X/≈ → A), f' ∘ η ＝ f e = (f' , r) , c
As mentioned above, if one so wishes, it is possible to resize down
the quotient X/≈ to the same universe as the given type X lives by
assuming propositional resizing. But we don’t see any mathematical
need or benefit to do so, as the constructed quotient, regardless of the universe
it inhabits, has a universal property that eliminates into any
desired universe, lower, equal or higher than the quotiented type.
The following axioms are together consistent by considering Voevodsky’s simplicial-set model:
We have that:
The first four admit a constructive interpretation via cubical type theory with an implementation in cubical Agda.
Univalence implies function extensionality and propositional extensionality.
Simple unique choice just holds.
Full unique choice is equivalent to function extensionality.
Choice implies excluded middle, as usual, and both are non-constructive.
Excluded middle implies propositional resizing and impredicativity.
The constructive status of propositional resizing and impredicativity is open.
Function extensionality and propositional resizing imply the existence of propositional truncations, and hence so do function extensionality and excluded middle.
The avoidance of excluded middle and choice makes the theory not only constructive but also applicable to more models. However, one is free to assume excluded middle and choice for pieces of mathematics that require them, or just if one simply prefers classical reasoning.
Univalent foundations have enough room for the constructive, non-constructive, pluralistic and neutral approaches to mathematics, and in this sense they are no different from e.g. set theoretic foundations.
An example of a theorem of univalent mathematics that requires excluded middle is Cantor-Schröder-Bernstein for homotopy types, or ∞-groupoids, which is also discussed in this blog post and is implemented in Agda.
A major omission in these notes is a discussion of higher-inductive types. On the other hand, these notes completely cover the foundational principles officially supported by UniMath, namely (1)-(7) above.
module ℕ-order-exercise-solution where _≤'_ : ℕ → ℕ → 𝓤₀ ̇ _≤'_ = ℕ-iteration (ℕ → 𝓤₀ ̇ ) (λ y → 𝟙) (λ f → ℕ-recursion (𝓤₀ ̇ ) 𝟘 (λ y P → f y)) open ℕ-order ≤-and-≤'-coincide : (x y : ℕ) → (x ≤ y) ＝ (x ≤' y) ≤-and-≤'-coincide 0 y = refl _ ≤-and-≤'-coincide (succ x) 0 = refl _ ≤-and-≤'-coincide (succ x) (succ y) = ≤-and-≤'-coincide x y module ℕ-more where open Arithmetic renaming (_+_ to _∔_) open basic-arithmetic-and-order ≤-prop-valued : (x y : ℕ) → is-subsingleton (x ≤ y) ≤-prop-valued 0 y = 𝟙-is-subsingleton ≤-prop-valued (succ x) zero = 𝟘-is-subsingleton ≤-prop-valued (succ x) (succ y) = ≤-prop-valued x y ≼-prop-valued : (x y : ℕ) → is-subsingleton (x ≼ y) ≼-prop-valued x y (z , p) (z' , p') = γ where q : z ＝ z' q = +-lc x z z' (x ∔ z ＝⟨ p ⟩ y ＝⟨ p' ⁻¹ ⟩ x ∔ z' ∎) γ : z , p ＝ z' , p' γ = to-subtype-＝ (λ z → ℕ-is-set (x ∔ z) y) q ≤-charac : propext 𝓤₀ → (x y : ℕ) → (x ≤ y) ＝ (x ≼ y) ≤-charac pe x y = pe (≤-prop-valued x y) (≼-prop-valued x y) (≤-gives-≼ x y) (≼-gives-≤ x y) the-subsingletons-are-the-subtypes-of-a-singleton : (X : 𝓤 ̇ ) → is-subsingleton X ⇔ (X ↪ 𝟙) the-subsingletons-are-the-subtypes-of-a-singleton X = φ , ψ where i : is-subsingleton X → is-embedding (!𝟙' X) i s ⋆ (x , refl ⋆) (y , refl ⋆) = ap (λ - → - , refl ⋆) (s x y) φ : is-subsingleton X → X ↪ 𝟙 φ s = !𝟙 , i s ψ : X ↪ 𝟙 → is-subsingleton X ψ (f , e) x y = d where a : x ＝ y → f x ＝ f y a = ap f {x} {y} b : is-equiv a b = embedding-gives-ap-is-equiv f e x y c : f x ＝ f y c = 𝟙-is-subsingleton (f x) (f y) d : x ＝ y d = inverse a b c the-subsingletons-are-the-subtypes-of-a-singleton' : propext 𝓤 → global-dfunext → (X : 𝓤 ̇ ) → is-subsingleton X ＝ (X ↪ 𝟙) the-subsingletons-are-the-subtypes-of-a-singleton' pe fe X = γ where a : is-subsingleton X ⇔ (X ↪ 𝟙) a = the-subsingletons-are-the-subtypes-of-a-singleton X b : is-subsingleton (X ↪ 𝟙) b (f , e) (f' , e') = to-subtype-＝ (being-embedding-is-subsingleton fe) (fe (λ x → 𝟙-is-subsingleton (f x) (f' x))) γ : is-subsingleton X ＝ (X ↪ 𝟙) γ = pe (being-subsingleton-is-subsingleton fe) b (pr₁ a) (pr₂ a) 𝔾↑-≃-equation : (ua : is-univalent (𝓤 ⊔ 𝓥)) → (X : 𝓤 ̇ ) → (A : (Σ Y ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) → 𝓦 ̇ ) → (a : A (Lift 𝓥 X , ≃-Lift X)) → 𝔾↑-≃ ua X A a (Lift 𝓥 X) (≃-Lift X) ＝ a 𝔾↑-≃-equation {𝓤} {𝓥} {𝓦} ua X A a = 𝔾↑-≃ ua X A a (Lift 𝓥 X) (≃-Lift X) ＝⟨ refl (transport A p a) ⟩ transport A p a ＝⟨ ap (λ - → transport A - a) q ⟩ transport A (refl t) a ＝⟨ refl a ⟩ a ∎ where t : (Σ Y ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) t = (Lift 𝓥 X , ≃-Lift X) p : t ＝ t p = univalence→'' {𝓤} {𝓤 ⊔ 𝓥} ua X t t q : p ＝ refl t q = subsingletons-are-sets (Σ Y ꞉ 𝓤 ⊔ 𝓥 ̇ , X ≃ Y) (univalence→'' {𝓤} {𝓤 ⊔ 𝓥} ua X) t t p (refl t) ℍ↑-≃-equation : (ua : is-univalent (𝓤 ⊔ 𝓥)) → (X : 𝓤 ̇ ) → (A : (Y : 𝓤 ⊔ 𝓥 ̇ ) → X ≃ Y → 𝓦 ̇ ) → (a : A (Lift 𝓥 X) (≃-Lift X)) → ℍ↑-≃ ua X A a (Lift 𝓥 X) (≃-Lift X) ＝ a ℍ↑-≃-equation ua X A = 𝔾↑-≃-equation ua X (Σ-induction A) has-section-charac : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → ((y : Y) → Σ x ꞉ X , f x ＝ y) ≃ has-section f has-section-charac f = ΠΣ-distr-≃ retractions-into : 𝓤 ̇ → 𝓤 ⁺ ̇ retractions-into {𝓤} Y = Σ X ꞉ 𝓤 ̇ , Y ◁ X pointed-types : (𝓤 : Universe) → 𝓤 ⁺ ̇ pointed-types 𝓤 = Σ X ꞉ 𝓤 ̇ , X retraction-classifier : Univalence → (Y : 𝓤 ̇ ) → retractions-into Y ≃ (Y → pointed-types 𝓤) retraction-classifier {𝓤} ua Y = retractions-into Y ≃⟨ i ⟩ (Σ X ꞉ 𝓤 ̇ , Σ f ꞉ (X → Y) , ((y : Y) → Σ x ꞉ X , f x ＝ y)) ≃⟨ ii ⟩ ((𝓤 /[ id ] Y)) ≃⟨ iii ⟩ (Y → pointed-types 𝓤) ■ where i = ≃-sym (Σ-cong (λ X → Σ-cong (λ f → ΠΣ-distr-≃))) ii = Id→Eq _ _ (refl _) iii = special-map-classifier (ua 𝓤) (univalence-gives-dfunext' (ua 𝓤) (ua (𝓤 ⁺))) id Y module surjection-classifier (pt : subsingleton-truncations-exist) (ua : Univalence) where hfe : global-hfunext hfe = univalence-gives-global-hfunext ua open basic-truncation-development pt hfe public _↠_ : 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ X ↠ Y = Σ f ꞉ (X → Y), is-surjection f surjections-into : 𝓤 ̇ → 𝓤 ⁺ ̇ surjections-into {𝓤} Y = Σ X ꞉ 𝓤 ̇ , X ↠ Y inhabited-types : (𝓤 : Universe) → 𝓤 ⁺ ̇ inhabited-types 𝓤 = Σ X ꞉ 𝓤 ̇ , ∥ X ∥ surjection-classifier : Univalence → (Y : 𝓤 ̇ ) → surjections-into Y ≃ (Y → inhabited-types 𝓤) surjection-classifier {𝓤} ua = special-map-classifier (ua 𝓤) (univalence-gives-dfunext' (ua 𝓤) (ua (𝓤 ⁺))) ∥_∥
Solutions are available at the end.
Exercise. A sequence of elements of a type X is just a function ℕ → X.
Use Cantor’s diagonal
argument
to show in Agda that the type of sequences of natural numbers is
uncountable. Prove a positive version and then derive a negative
version from it:
positive-cantors-diagonal : (e : ℕ → (ℕ → ℕ)) → Σ α ꞉ (ℕ → ℕ), ((n : ℕ) → α ≠ e n) cantors-diagonal : ¬(Σ e ꞉ (ℕ → (ℕ → ℕ)) , ((α : ℕ → ℕ) → Σ n ꞉ ℕ , α ＝ e n))
Hint. It may be helpful to prove that the function succ has no
fixed points, first.
Exercise.
𝟚-has-𝟚-automorphisms : dfunext 𝓤₀ 𝓤₀ → (𝟚 ≃ 𝟚) ≃ 𝟚
Now we would like to have (𝟚 ＝ 𝟚) ＝ 𝟚 with univalence, but the
problem is that the type 𝟚 ＝ 𝟚 lives in 𝓤₁ whereas 𝟚 lives in
𝓤₀ and so, having different types, can’t be compared for equality.
But we do have that
lifttwo : is-univalent 𝓤₀ → is-univalent 𝓤₁ → (𝟚 ＝ 𝟚) ＝ Lift 𝓤₁ 𝟚
Exercise. Having decidable equality is a subsingleton type.
Exercise. We now discuss alternative formulations of the principle of excluded middle.
DNE : ∀ 𝓤 → 𝓤 ⁺ ̇ DNE 𝓤 = (P : 𝓤 ̇ ) → is-subsingleton P → ¬¬ P → P ne : (X : 𝓤 ̇ ) → ¬¬(X + ¬ X) DNE-gives-EM : dfunext 𝓤 𝓤₀ → DNE 𝓤 → EM 𝓤 EM-gives-DNE : EM 𝓤 → DNE 𝓤
The following says that excluded middle holds if and only if every subsingleton is the negation of some type.
SN : ∀ 𝓤 → 𝓤 ⁺ ̇ SN 𝓤 = (P : 𝓤 ̇ ) → is-subsingleton P → Σ X ꞉ 𝓤 ̇ , P ⇔ ¬ X SN-gives-DNE : SN 𝓤 → DNE 𝓤 DNE-gives-SN : DNE 𝓤 → SN 𝓤
succ-no-fixed-point : (n : ℕ) → succ n ≠ n succ-no-fixed-point 0 = positive-not-zero 0 succ-no-fixed-point (succ n) = γ where IH : succ n ≠ n IH = succ-no-fixed-point n γ : succ (succ n) ≠ succ n γ p = IH (succ-lc p) positive-cantors-diagonal = sol where sol : (e : ℕ → (ℕ → ℕ)) → Σ α ꞉ (ℕ → ℕ), ((n : ℕ) → α ≠ e n) sol e = (α , φ) where α : ℕ → ℕ α n = succ(e n n) φ : (n : ℕ) → α ≠ e n φ n p = succ-no-fixed-point (e n n) q where q = succ (e n n) ＝⟨ refl (α n) ⟩ α n ＝⟨ ap (λ - → - n) p ⟩ e n n ∎ cantors-diagonal = sol where sol : ¬(Σ e ꞉ (ℕ → (ℕ → ℕ)) , ((α : ℕ → ℕ) → Σ n ꞉ ℕ , α ＝ e n)) sol (e , γ) = c where α : ℕ → ℕ α = pr₁ (positive-cantors-diagonal e) φ : (n : ℕ) → α ≠ e n φ = pr₂ (positive-cantors-diagonal e) b : Σ n ꞉ ℕ , α ＝ e n b = γ α c : 𝟘 c = φ (pr₁ b) (pr₂ b) 𝟚-has-𝟚-automorphisms = sol where sol : dfunext 𝓤₀ 𝓤₀ → (𝟚 ≃ 𝟚) ≃ 𝟚 sol fe = invertibility-gives-≃ f (g , η , ε) where f : (𝟚 ≃ 𝟚) → 𝟚 f (h , e) = h ₀ g : 𝟚 → (𝟚 ≃ 𝟚) g ₀ = id , id-is-equiv 𝟚 g ₁ = swap₂ , swap₂-is-equiv η : (e : 𝟚 ≃ 𝟚) → g (f e) ＝ e η (h , e) = γ (h ₀) (h ₁) (refl (h ₀)) (refl (h ₁)) where γ : (m n : 𝟚) → h ₀ ＝ m → h ₁ ＝ n → g (h ₀) ＝ (h , e) γ ₀ ₀ p q = !𝟘 (g (h ₀) ＝ (h , e)) (₁-is-not-₀ (equivs-are-lc h e (h ₁ ＝⟨ q ⟩ ₀ ＝⟨ p ⁻¹ ⟩ h ₀ ∎))) γ ₀ ₁ p q = to-subtype-＝ (being-equiv-is-subsingleton fe fe) (fe (𝟚-induction (λ n → pr₁ (g (h ₀)) n ＝ h n) (pr₁ (g (h ₀)) ₀ ＝⟨ ap (λ - → pr₁ (g -) ₀) p ⟩ pr₁ (g ₀) ₀ ＝⟨ refl ₀ ⟩ ₀ ＝⟨ p ⁻¹ ⟩ h ₀ ∎) (pr₁ (g (h ₀)) ₁ ＝⟨ ap (λ - → pr₁ (g -) ₁) p ⟩ pr₁ (g ₀) ₁ ＝⟨ refl ₁ ⟩ ₁ ＝⟨ q ⁻¹ ⟩ h ₁ ∎))) γ ₁ ₀ p q = to-subtype-＝ (being-equiv-is-subsingleton fe fe) (fe (𝟚-induction (λ n → pr₁ (g (h ₀)) n ＝ h n) (pr₁ (g (h ₀)) ₀ ＝⟨ ap (λ - → pr₁ (g -) ₀) p ⟩ pr₁ (g ₁) ₀ ＝⟨ refl ₁ ⟩ ₁ ＝⟨ p ⁻¹ ⟩ h ₀ ∎) (pr₁ (g (h ₀)) ₁ ＝⟨ ap (λ - → pr₁ (g -) ₁) p ⟩ pr₁ (g ₁) ₁ ＝⟨ refl ₀ ⟩ ₀ ＝⟨ q ⁻¹ ⟩ h ₁ ∎))) γ ₁ ₁ p q = !𝟘 (g (h ₀) ＝ (h , e)) (₁-is-not-₀ (equivs-are-lc h e (h ₁ ＝⟨ q ⟩ ₁ ＝⟨ p ⁻¹ ⟩ h ₀ ∎))) ε : (n : 𝟚) → f (g n) ＝ n ε ₀ = refl ₀ ε ₁ = refl ₁ lifttwo = sol where sol : is-univalent 𝓤₀ → is-univalent 𝓤₁ → (𝟚 ＝ 𝟚) ＝ Lift 𝓤₁ 𝟚 sol ua₀ ua₁ = Eq→Id ua₁ (𝟚 ＝ 𝟚) (Lift 𝓤₁ 𝟚) e where e = (𝟚 ＝ 𝟚) ≃⟨ Id→Eq 𝟚 𝟚 , ua₀ 𝟚 𝟚 ⟩ (𝟚 ≃ 𝟚) ≃⟨ 𝟚-has-𝟚-automorphisms (univalence-gives-dfunext ua₀) ⟩ 𝟚 ≃⟨ ≃-sym (Lift-≃ 𝟚) ⟩ Lift 𝓤₁ 𝟚 ■ hde-is-subsingleton : dfunext 𝓤 𝓤₀ → dfunext 𝓤 𝓤 → (X : 𝓤 ̇ ) → is-subsingleton (has-decidable-equality X) hde-is-subsingleton fe₀ fe X h h' = c h h' where a : (x y : X) → is-subsingleton (decidable (x ＝ y)) a x y = +-is-subsingleton' fe₀ b where b : is-subsingleton (x ＝ y) b = hedberg h x y c : is-subsingleton (has-decidable-equality X) c = Π-is-subsingleton fe (λ x → Π-is-subsingleton fe (a x)) ne = sol where sol : (X : 𝓤 ̇ ) → ¬¬(X + ¬ X) sol X = λ (f : ¬(X + ¬ X)) → f (inr (λ (x : X) → f (inl x))) DNE-gives-EM = sol where sol : dfunext 𝓤 𝓤₀ → DNE 𝓤 → EM 𝓤 sol fe dne P i = dne (P + ¬ P) (+-is-subsingleton' fe i) (ne P) EM-gives-DNE = sol where sol : EM 𝓤 → DNE 𝓤 sol em P i = γ (em P i) where γ : P + ¬ P → ¬¬ P → P γ (inl p) φ = p γ (inr n) φ = !𝟘 P (φ n) SN-gives-DNE = sol where sol : SN 𝓤 → DNE 𝓤 sol {𝓤} sn P i = h where X : 𝓤 ̇ X = pr₁ (sn P i) f : P → ¬ X f = pr₁ (pr₂ (sn P i)) g : ¬ X → P g = pr₂ (pr₂ (sn P i)) f' : ¬¬ P → ¬(¬¬ X) f' = contrapositive (contrapositive f) h : ¬¬ P → P h = g ∘ tno X ∘ f' h' : ¬¬ P → P h' φ = g (λ (x : X) → φ (λ (p : P) → f p x)) DNE-gives-SN = sol where sol : DNE 𝓤 → SN 𝓤 sol dne P i = ¬ P , dni P , dne P i
Without the following list of operator precedences and associativities (left or right), this agda file doesn’t parse and is rejected by Agda.
infix 0 _∼_ infixr 50 _,_ infixr 30 _×_ infixr 20 _+_ infixl 70 _∘_ infix 0 Id infix 0 _＝_ infix 10 _⇔_ infixl 30 _∙_ infixr 0 _＝⟨_⟩_ infix 1 _∎ infix 40 _⁻¹ infix 10 _◁_ infixr 0 _◁⟨_⟩_ infix 1 _◀ infix 10 _≃_ infixl 30 _●_ infixr 0 _≃⟨_⟩_ infix 1 _■ infix 40 _∈_ infix 30 _[_,_] infixr -1 -Σ infixr -1 -Π infixr -1 -∃!