On a (terminally connected, pro-etale) factorization of geometric morphisms
Olivia Caramello, Axel Osmond
TL;DR
The paper generalizes the classical (connected, etale) factorization to a (terminally_connected, pro-etale) factorization that applies to all geometric morphisms between Grothendieck topoi. It defines pro-etale morphisms as cofiltered bilimits of etale morphisms and shows every morphism factors as a terminally connected morphism followed by a pro-etale one, with a canonical presentation via global elements. It provides intrinsic characterizations, shows pro-etale morphisms are localic and discrete opfibrations, and develops laxness and stability properties that mirror 2-dimensional factorization systems. The results yield a robust framework for understanding how global elements and fibers drive factorization and (co)limits in the topos-theoretic setting, with connections to germs, localization, and initiality phenomena.
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.