Local fibrations and morphisms of relative toposes

TL;DR

This paper introduces local fibrations, a generalization of fibrations considering Grothendieck topologies, and explores their role in morphisms of relative toposes, including characterizations and a version of Diaconescu's theorem.

Contribution

It defines local fibrations, extends classical fibrations to this setting, and characterizes morphisms of relative sites and toposes using this new concept.

Findings

Characterization of functors inducing morphisms between relative toposes

Weak version of Diaconescu's theorem for local fibrations

Fibrational characterization of locally and totally connected morphisms

Abstract

We introduce the notion of local fibration, a generalization of the notion of fibration which takes into account the presence of Grothendieck topologies on the two categories, and show that the classical results about fibrations lift to this more general setting. As an application of this notion, we obtain a characterization of the functors between relative sites that induce a morphism between the corresponding relative toposes: these are exactly the morphisms of sites which are morphisms of local fibrations. Also, we prove a weak version of Diaconescu's theorem, providing an equivalence between the continuous morphisms of local fibrations towards the canonical stack of a relative topos and the weak morphisms between the associated indexed toposes. The paper also contains a number of results of independent interest on morphisms of toposes and their associated stacks, including a fibrational characterization of locally connected (resp. totally connected) morphisms.