Martin Escardo

@EscardoMartin

In the paper "Continuous lattices", Dana Scott did two things. After defining continuous lattices, he proved that the injective topological spaces are precisely the continuous lattices endowed with the Scott topology. 2/

13/06/2021, 21:17:47

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Martin Escardo

@EscardoMartin

The second thing he did was to give a model of the untyped lambda calculus with types interpreted as continuous lattices. 3/

13/06/2021, 21:18:46

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Martin Escardo

@EscardoMartin

Just before this paper, Brian Day and Max Kelly used continuous lattices before a definition of continuous lattice was known. They proved that topological space is exponentiable if and only if its lattice of open sets is continuous. 4/

13/06/2021, 21:21:10

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Martin Escardo

@EscardoMartin

But even before that, Jimmie Lawson was looking at compact Hausdorff semilattices (so topological algebra). Fast forwarding, it turned out that what we was staring at was continuous lattices under Lawson topology. 5/

13/06/2021, 21:23:29

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Martin Escardo

@EscardoMartin

The Scott and Lawson topologies on continuous lattice relate in a very interesting way. The Lawson topology is the patch of the Scott topology, where the patch is a categorical correflection into compact Hausdorff spaces. 6/

13/06/2021, 21:26:06

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Martin Escardo

@EscardoMartin

Regarding Stone duality, we have that the distributive continuous lattices are precisely (up to isomorphism, of course) the lattices of open sets of sober locally compact spaces. 7/

13/06/2021, 21:27:32

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Martin Escardo

@EscardoMartin

I am told by the authors of the "Compendium of continuous lattices" (the predecessor of "Continuous lattices and domains) that Dana Scott (one of the coauthors) never told them, at that time, what the connection with computation was. 8/

13/06/2021, 21:29:46

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Martin Escardo

@EscardoMartin

Jimmie Lawson told me a few years ago that it was an incredible coincidence, at that time (around 1965-1975) that not only several people came across the concept but that they actually met and developed in much further. 9/

13/06/2021, 21:32:11

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Martin Escardo

@EscardoMartin

Another Day, namely Alan Day, showed that the continuous lattices are precisely the algebras of the filter monad. There are actually two filter monads: on Set and on Top. The answer works for both.

13/06/2021, 21:34:37

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Martin Escardo

@EscardoMartin

And the story goes on and on. 11/11.
NB. I forgot to number 10/11, the previous one.

13/06/2021, 21:36:14

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Martin Escardo

@EscardoMartin

Bibliography.
(1) A compendium of continuous lattices, 1980. https://www.springer.com/gp/book/9783642676802
(2) Continuous lattices and domains, 2009. https://www.cambridge.org/core/books/continuous-lattices-and-domains/D27E855CD9C88E6DB3C98196B88F0AF6#
The authors of (1) and (2) are G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott.

13/06/2021, 22:40:37

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Martin Escardo

@EscardoMartin

NB. (2) is an extended version of (1), with some errors fixed. In particular, domains more general than continuous lattices, which occur in applications to computer science, are included.

14/06/2021, 11:51:04

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