Handbook of Categorical Algebra: Volume 3, Sheaf Theory
Cambridge University Press, 8 déc. 1994 - 522 pages
The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.
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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