Abstract
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.
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© 1994 Springer Science+Business Media New York
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Mac Lane, S., Moerdijk, I. (1994). Geometric Morphisms. In: Sheaves in Geometry and Logic. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0927-0_9
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DOI: https://doi.org/10.1007/978-1-4612-0927-0_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97710-2
Online ISBN: 978-1-4612-0927-0
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