Geometric morphisms between toposes of monoid actions: factorization systems

TL;DR

This paper explores how geometric morphisms between toposes of monoid actions relate to properties of monoid homomorphisms, providing a systematic analysis and applications to topos-theoretic Galois theory.

Contribution

It establishes correspondences between geometric morphism properties and monoid homomorphism properties, especially within factorization systems, and applies these results to Galois theory.

Findings

Characterization of geometric morphisms via monoid homomorphisms

Analysis of factorization systems in toposes of monoid actions

Application to topos-theoretic Galois theory

Abstract

Let M, N be monoids, and PSh(M), PSh(N) their respective categories of right actions on sets. In this paper, we systematically investigate correspondences between properties of geometric morphisms PSh(M) $\rightarrow$ PSh(N) and properties of the semigroup homomorphisms M $\rightarrow$ N or flat-left-N-right-M-sets inducing them. More specifically, we consider properties of geometric morphisms featuring in factorization systems, namely: surjections, inclusions, localic morphisms, hyperconnected morphisms, terminal-connected morphisms, {\'e}tale morphisms, pure morphisms and complete spreads. We end with an application to topos-theoretic Galois theory to the special case of toposes of the form PSh(M).