Handbook of categorical algebra. 3: Categories of sheaves. (English) Zbl 0911.18001

Borceux, Francis

Encyclopedia of Mathematics and Its Applications. 52. Cambridge: Cambridge Univ. Press. xvii, 522 p. (1994).

The third volume [F. Borceux, Vol. 1: Basic category theory (1994; Zbl 0803.18001), Vol. 2: Categories and structures (1994; Zbl 0843.18001)] is entirely devoted to the study of categories of sheaves: sheaves on a space, a locale, a site. This is the opportunity for developing the essential aspects of the theory of locales and introducing Grothendieck toposes. We relate this with the algebraic aspects of volume 2 by proving in this context the existence of a classifying topos for coherent theories. All these considerations lead naturally to the notion of an elementary topos. We study quite extensively the internal logic of toposes, including the law of excluded middle and the axiom of infinity. We conclude by showing how toposes are a natural context for defining sheaves.

MSC:

18-00	General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to category theory
03G30	Categorical logic, topoi
18B25	Topoi
18F10	Grothendieck topologies and Grothendieck topoi

Keywords:

categories of sheaves; locales; Grothendieck toposes; classifying topos for coherent theories; internal logic of toposes; law of excluded middle; axiom of infinity

Citations:

Zbl 0803.18001; Zbl 0843.18001