Geometric Morphisms

  • Saunders Mac Lane
  • Ieke Moerdijk
Part of the Universitext book series (UTX)


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 XY 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 FEε of topoi correspond to Lawvere-Tierney topologies in the codomain ε, while surjections FE correspond to left exact comonads on the domain.F.


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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Saunders Mac Lane
    • 1
  • Ieke Moerdijk
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Mathematical InstituteUniversity of UtrechtUtrechtThe Netherlands

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