April 16, Freek Geerligs, Synthetic Stone duality

[Dan Christensen]

This week the HoTTEST seminar presents:

Freek Geerligs

Synthetic Stone duality

The talk is at 11:30am EDT (15:30 UTC) on Thursday, April 16. The talk will be 60 minutes long, followed by up to 30 minutes for questions. See https://hottest-seminar.github.io/ for the Zoom link and a list of all upcoming talks.

All are welcome!

Abstract:

In this talk, we will give an overview of Synthetic Stone duality (https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.3). We will then discuss some related work in progress.

Synthetic Stone duality is an extension of homotopy type theory with four axioms. These axioms are strong enough to decide Bishop's omniscience principles. We introduce a (synthetic) topology on any type, such that all functions are continuous. We are interested in Stone spaces and compact Hausdorff spaces, where the topology behaves as one would expect. In particular, we can define the (topological) interval and show that all functions are continuous in the epsilon-delta sense.

Currently, we are working on a paper with a method for calculating cohomology with countably presented coefficients for compact Hausdorff spaces. We are also interested in a correspondence between homotopical concepts defined using traditional topology (using paths from the topological interval) and homotopy type theory (using identity types).

This talk will contain joint work with Reid Barton, Felix Cherubini, Thierry Coquand, and Hugo Moeneclaey.