Titlebook: Sheaves in Geometry and Logic; A First Introduction Saunders Mac Lane,Ieke Moerdijk Textbook 1994 Springer-Verlag New York, Inc. 1994 Algeb

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Geometric Logic and Classifying Topoi,Many mathematical structures can be axiomatized by formulas of this form (1).For instance, local rings are axiomatized by the usual equations for a commutative ring with unit, together with the axiom. which states that the ring is local; this axiom (2) is indeed of the form (1).

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Localic Topoi, the study of such locales, motivated by the topological examples. It will turn out that a topological space is essentially determined by its lattice of open sets when that space . has the property of “sobriety”, but, beyond that point, spaces and locales diverge.

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Prologue, F. W. Lawvere in his search for an axiomatization of the category of sets and that of “variable” sets, and with Paul Cohen in the use of forcing to construct new models of Zermelo-Frwnkel set theory.

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Categories of Functors,ies can be efficiently developed. Superficially quite different categories, arising in geometry, topology, algebraic geometry, group representations, and set theory, all turn out to satisfy the axioms defining such a topos.

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Categorical Preliminaries,ll be familiar with these preliminaries; they should immediately start with Chapter I, referring back to these preliminaries whenever necessary. On the other hand, these preliminaries do not present sufficiently many examples and are by no means enough to constitute a proper introduction to category

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