Denseness conditions, morphisms and equivalences of toposes

TL;DR

This paper provides a comprehensive analysis of topos morphisms and equivalences, introducing new concepts and characterizations that deepen understanding of topos theory and its morphisms.

Contribution

It introduces the notion of weak morphisms of toposes, characterizes when morphisms induce equivalences, and offers site characterizations of various topos morphism properties.

Findings

Characterization of equivalences of toposes via site conditions

Introduction of weak morphisms and their properties

Site descriptions of various geometric morphism factorizations

Abstract

We systematically investigate morphisms and equivalences of toposes from multiple points of view. We establish a dual adjunction between morphisms and comorphisms of sites, introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms. In particular, we examine continuous comorphism of sites and show that this class of comorphisms notably includes all fibrations as well as morphisms of fibrations. We also establish a characterization theorem for essential geometric morphisms and locally connected morphisms in terms of continuous functors, and a relative version of the comprehensive factorization of a functor. Then we prove a general theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes. This stems from a detailed analysis of arrows in Grothendieck toposes and denseness conditions, which yields results of independent interest. We also derive site characterizations of the property of a geometric morphism to be an inclusion (resp. a surjection, hyperconnected, localic), as well as site-level descriptions of the surjection-inclusion and hyperconnected-localic factorizations of a geometric morphism.