Stability of properties of locales under groups

TL;DR

This paper demonstrates that certain categorical properties of locales are preserved under the formation of categories of G-objects for internal groups, extending to groupoids, with implications for categorical axioms and topos theory.

Contribution

It proves that properties of locales satisfying specific axioms are maintained when forming categories of G-objects for internal groups, including non-exponentiable cases, and discusses extensions to groupoids.

Findings

Properties of locales are stable under G-object constructions.

The categorical axioms are preserved in categories of G-objects.

An example shows the axioms do not imply the existence of an elementary topos.

Abstract

Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if $\mathcal{C}$ is a category that satisfies the axioms then so too is the category $[ G, \mathcal{C}]$ of $G$-objects, for any internal group $G$. To achieve this we prove a general categorical result: if an object $S$ is double exponentiable in a category with finite products then so is its associated trivial $G$-object $(S, \pi_2: G \times S \rightarrow S)$. The result holds even if $S$ is not exponentiable. An example is given of a category $\mathcal{C}$ that satisfies the axioms, but for which there is no elementary topos $\mathcal{E}$ such that $\mathcal{C}$ is the category of locales over $\mathcal{E}$. It is shown, in outline, how the results can be extended from groups to groupoids.