A Foundation for Synthetic Stone Duality

TL;DR

This paper develops a synthetic topology framework within homotopy type theory for light condensed sets, enabling internal reasoning about classical topological results like Brouwer's fixed-point theorem.

Contribution

It extends homotopy type theory with new axioms to model synthetic topology for light condensed sets, proving classical theorems internally.

Findings

All functions between certain types are continuous.

The topology on types matches classical expectations.

Brouwer's fixed-point theorem is proved internally.

Abstract

The language of homotopy type theory has proved to be appropriate as an internal language for various higher toposes, for example with Synthetic Algebraic Geometry for the Zariski topos. In this paper we apply such techniques to the higher topos corresponding to the light condensed sets of Dustin Clausen and Peter Scholze. This seems to be an appropriate setting to develop synthetic topology, similar to the work of Mart\'in Escard\'o. To reason internally about light condensed sets, we use homotopy type theory extended with 4 axioms. Our axioms are strong enough to prove Markov's principle, LLPO and the negation of WLPO. We also define a type of open propositions, inducing a topology on any type. This leads to a synthetic topological study of (second countable) Stone and compact Hausdorff spaces. Indeed all functions are continuous in the sense that they respect this induced topology, and this topology is as expected for these classes of types. For example, any map from the unit interval to itself is continuous in the usual epsilon-delta sense. We also use the synthetic homotopy theory given by the higher types of homotopy type theory to define and work with cohomology. As an application, we prove Brouwer's fixed-point theorem internally.