A topos-theoretic approach to Stone-type dualities

TL;DR

This paper introduces a topos-theoretic framework that unifies and extends various Stone-type dualities, enabling the derivation of new dualities and connecting properties across preorders, locales, and topological spaces.

Contribution

It provides a unifying topos-theoretic approach to Stone dualities, generating new dualities and linking properties of preorders with their associated spaces and models.

Findings

Multiple known dualities are unified under a single topos-theoretic framework.

New dualities between preordered structures and topological spaces are systematically generated.

The approach connects properties of preorders with those of locales and topological spaces.

Abstract

We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new dualities between preordered structures and locales or topological spaces can be generated through our topos-theoretic machinery in a uniform way. We then apply our topos-theoretic interpetation to obtain results connecting properties of preorders and properties of the corresponding locales or topological spaces, and we establish adjunctions between various kinds of categories as natural applications of our general methodology. In the last part of the paper, we exploit the theory developed in the previous parts to obtain a topos-theoretic interpretation of the problem of finding explicit descriptions of models of 'ordered algebraic theories' presented by generators and relations, and give several examples which illustrate the effectiveness of our methodology. In passing, we provide a number of other applications of our theory to Algebra, Topology and Logic.