A Foundation for Synthetic Stone Duality

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.