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Theories, Sites, Toposes

Olivia Caramello
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Felipe Zaldivar
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In the expanding universe of generalizations of a topological space, Grothendieck’s notion of a site isolates and axiomatizes the notion of an open cover of a space, allowing the inclusions to become any family of morphisms with some basic properties (e.g., closed under composition and base change). These Grothendieck topologies allow generalizations of the notion of sheaf and hence of sheaf cohomologies, which are the Weil cohomologies which were used to formulate, and eventually prove, the Weil conjectures on rational points of varieties over finite fields.

The construction and main properties of the category of sheaves on a site were rapidly recognized as purely categorical, and the notion of a Grothendieck topos captures this categorical formalism. Hence, all sites are examples of toposes, including the category of sets, where a set is now a sheaf on a point.

Along the way, the notion of point of a geometric object also evolved. In the new formalism, a point in a given topos is a geometric morphism from the topos of sets to the given topos. Here, a geometric morphism between toposes is a pair of adjoint functors, where the left adjoint preserves finite limits. There are examples of toposes with no points, which has prompted the quip pointless topology.

Grothendieck toposes are rooted in arithmetic algebraic geometry, but since the category of sets is now just one example of a topos, choosing it for the usual set-theoretical based mathematics is just an inherited tradition or prejudice. It didn’t take long for mathematical logicians to distill a more general notion of elementary topos. As a result, several versions of logic depend just in the choice of elementary topos, even logics with no excluded middle. First order logic is then interpreted in certain categories, and geometric categories and geometric functors between them are then defined. In particular, every Grothendieck topos is a geometric category.

First order geometric theories are embodied in the theory of classifying toposes. A fundamental fact is that every geometric theory admits a classifying topos and every Grothendieck topos is the classifying topos of some geometric theory.

From W. Lawvere’s thesis, Functorial Semantics of Algebraic Theories (Columbia, 1963), to the Montreal school of G. Reyes, M. Makkai, A. Joyal, categorical logic flourished in the 1970s, but the notion of a classifying topos somehow stagnated. The main goal of the book under review is to systematically recall and develop the theory of classifying toposes, focusing on some particular important problems where this notion plays a crucial role. Along the way, and with the aim of providing an answer to those questions, the author systematizes her unifying “toposes as bridges” technique, which transfers information between different geometric theories with the same classifying topos. Here, by “information” one means topos-theoretical invariants or categorical constructions stable under categorical equivalence.

The book is systematic, developing the subject from its very categorical beginning, assuming just the basic notions of category theory and a familiarity with first-order logic. This is a research monograph, but a dedicated reader would certainly profit from it. Although the categorical setting is a little daunting, the concrete applications of the author’s methods range from geometric logic to Nori motives, if one allows that these subjects are down-to-earth.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is

1. Topos-theoretic background
2. Classifying toposes and the 'bridge' technique
3. A duality theorem
4. Lattices of theories
5. Flat functors and classifying toposes
6. Theories of presheaf type: general criteria
7. Expansions and faithful interpretations
8. Quotients of a theory of presheaf type
9. Examples of theories of presheaf type
10. Some applications