Handbook of Categorical Algebra: Volume 3, Sheaf Theory

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. The last five chapters are then devoted to an axiomatic approach to the categories of sheaves, via the theory of elementary toposes. First are established the exactness properties of toposes. A long chapter is devoted to a careful, accessible and extensive description of the internal logic of toposes, a very powerful tool. The book ends by looking at three specific topics: the natural number object in a topos, the Boolean toposes and the theory of sheaves in an arbitrary topos.