Handbook of Categorical Algebra: Volume 3, Sheaf Theory
Cambridge University Press, 8 déc. 1994 - 522 pages
The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. This third volume turns to topos theory and the idea of sheaves. The theory of locales is considered first, and Grothendieck toposes are introduced. Notions of sketchability and accessible categories are discussed, and an axiomatic generalization of the category of sheaves is given.There is ample material here for a graduate course in category theory, and the book should also serves as a reference for users.
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adjunction associated sheaf functor axiom bijection cartesian closedness category of presheaves category of sheaves characteristic morphism classifying topos coherent formula colimit composite conditions are equivalent consider diagram coproduct Corollary corresponding deﬁned deﬁnition epimorphic family epimorphism étale morphism exists factorization ﬁlter ﬁrst following conditions functor F G F(u G-sets geometric morphism given global element Grothendieck topology Grothendieck topos Heyting algebra implies isomorphism left adjoint Let F Let us prove monomorphism morphism f morphism of locales natural number object natural transformation notation observe open subsets poset presheaves Proposition pullback pullback of diagram Q-sets reﬂection remains to prove representable functors right adjoint satisﬁes set theory sheaf F singleton small category subobject subobject classiﬁer subpresheaf term of type Theorem topological space topos of sheaves toposes truth table variable of type volume yields Yoneda lemma