Artin glueings of toposes as adjoint split extensions

TL;DR

This paper generalizes the concept of Artin glueings from frames to toposes, establishing a 2-categorical framework that relates split extensions and morphisms in the context of topos theory.

Contribution

It extends the theory of Artin glueings to toposes and introduces a 2-categorical notion of split extensions, along with a category of such extensions and their equivalence to Hom functors.

Findings

Ext(H,N) is contravariantly equivalent to Hom(H,N)

The equivalence extends to a 2-natural contravariant equivalence

Artin glueings correspond to a 2-categorical notion of split extensions

Abstract

Artin glueings of frames correspond to adjoint split extensions in the category of frames and finite-meet-preserving maps. We extend these ideas to the setting of toposes and show that Artin glueings of toposes correspond to a 2-categorical notion of adjoint split extensions in the 2-category of toposes, finite-limit-preserving functors and natural transformations. A notion of morphism between these split extensions is introduced, which allows the category Ext(H,N) to be constructed. We show that Ext(H,N) is contravariantly equivalent to Hom(H,N), and moreover, that this can be extended to a 2-natural contravariant equivalence between the Hom 2-functor and a naturally defined Ext 2-functor.