Geometric Morphisms

In this chapter, we begin the study of the maps between topoi: the so-called geometric morphisms. The definition is modeled on the case of topological spaces, where a continuous map X→Y gives rise to an adjoint pair Sh(X) ⇄Sh(Y) of functors between sheaf topoi. The first two sections of this chapter are concerned mainly with a number of examples, and with the construction of the necessary adjunctions by analogues of the ®-Hom adjunction of module theory. In a third section, we consider two special types of geometric morphisms: the embeddings and the surjections. For these two types, there is a factorization theorem, parallel to the familiar factorization of a function as a surjection followed by an injection. Moreover, we prove that the embeddings F→Eε of topoi correspond to Lawvere-Tierney topologies in the codomain ε, while surjections F↠E correspond to left exact comonads on the domain.F.