On a (terminally connected, pro-etale) factorization of geometric morphisms

TL;DR

This paper generalizes the classical factorization of geometric morphisms to a broader class involving pro-etale morphisms, exploring their properties and stability within Grothendieck topoi.

Contribution

It introduces a (terminally connected, pro-etale) factorization for all geometric morphisms, extending classical results and analyzing their properties and stability.

Findings

Pro-etale morphisms relate closely to global elements of inverse images.

The paper establishes stability properties of pro-etale morphisms.

Discusses fibrational aspects of these morphisms.

Abstract

We extend the classical (connected, etale) factorization of locally connected geometric morphisms into a (terminally connected, pro-etale) factorization for all geometric morphisms between Grothendieck topoi. We discuss properties of both classes of morphisms, particularly the relation between pro-etale geometric morphisms and the category of global elements of their inverse image; we also discuss their stability properties as well as some fibrational aspects.