Handbook of Categorical Algebra: Volume 3, Sheaf TheoryCambridge 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. |
Table des matières
1 Locales | 1 |
2 Sheaves | 87 |
3 Grothendieck toposes | 192 |
4 The classifying topos | 244 |
5 Elementary toposes | 288 |
6 Internal logic of a topos | 342 |
7 The law of excluded middle | 432 |
8 The axiom of infinity | 453 |
9 Sheaves in a topos | 486 |
| 514 | |
| 517 | |
Expressions et termes fréquents
A₁ adjunction associated sheaf functor axiom b₁ bijection cartesian closedness category of presheaves category of sheaves characteristic morphism classifying topos coherent formula colimit composite conditions are equivalent coproduct Corollary corresponding defined definition epimorphic family epimorphism étale morphism exists factorization following conditions formula with free free variables x1,...,xn functor F G-sets geometric morphism given global element Grothendieck topology Grothendieck topos Heyting algebra implies inverse isomorphism left adjoint left exact functor Let us prove monomorphism morphism f morphism of locales N-sets natural number object natural transformation notation observe open subsets poset presheaves Proposition pullback pullbacks of diagram remains to prove representable functors right adjoint set theory sheaf F singleton small category Sp(R subobject classifier subpresheaf term of type Theorem topological space topos of sheaves toposes truth table types X1 variable of type volume yields Yoneda lemma ΩΑ