Toposes of Discrete Monoid Actions

TL;DR

This paper explores the properties and classifications of toposes arising from discrete monoid actions, establishing an equivalence between monoids and toposes through their canonical points.

Contribution

It provides a characterization of toposes of right monoid actions and solves the Morita equivalence problem via a 2-category equivalence.

Findings

Toposes of right M-sets are characterized by their canonical points.

An equivalence between 2-categories of monoids and toposes is established.

The Morita equivalence problem is solved in this context.

Abstract

Properties of toposes of right $M$-sets are studied, and these toposes are characterised up to equivalence by their canonical points. The solution to the corresponding Morita equivalence problem is presented in the form of an equivalence between a 2-category of monoids and the corresponding 2-category of toposes.