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Artin glueings of frames correspond to adjoint split extensions in thecategory of frames and finite-meet-preserving maps. We extend these ideas tothe setting of toposes and show that Artin glueings of toposes correspond to a2-categorical notion of adjoint split extensions in the 2-category of toposes,finite-limit-preserving functors and natural transformations. A notion ofmorphism between these split extensions is introduced, which allows thecategory Ext(H,N) to be constructed. We show that Ext(H,N) is contravariantlyequivalent to Hom(H,N), and moreover, that this can be extended to a 2-naturalcontravariant equivalence between the Hom 2-functor and a naturally defined Ext2-functor.

Artin glueings of toposes as adjoint split extensions