Toposes of Monoid Actions

TL;DR

This paper characterizes toposes arising from monoid actions, including both discrete and topological cases, by studying presheaf toposes and supercompactly generated toposes, providing criteria for their equivalence.

Contribution

It introduces a comprehensive framework for understanding toposes of monoid actions, extending from ordinary to topological monoids, with new characterizations and conditions.

Findings

Characterization of presheaf toposes from monoid actions

Introduction of supercompactly generated toposes

Necessary and sufficient conditions for toposes of topological monoid actions

Abstract

We study toposes of actions of monoids on sets. We begin with ordinary actions, producing a class of presheaf toposes which we characterize. As groundwork for considering topological monoids, we branch out into a study of supercompactly generated toposes (a class strictly larger than presheaf toposes). This enables us to efficiently study and characterize toposes of continuous actions of topological monoids on sets, where the latter are viewed as discrete spaces. Finally, we refine this characterization into necessary and sufficient conditions for a supercompactly generated topos to be equivalent to a topos of this form.