Theories, Sites, Toposes

Abstract

This book is devoted to a general study of geometric theories from a topos-theoretic perspective. After recalling the necessary topos-theoretic preliminaries, it presents the main methodology it uses to extract ‘concrete’ information on theories from properties of their classifying toposes—the ‘bridge’ technique. As a first implementation of this methodology, a duality is established between the subtoposes of the classifying topos of a geometric theory and the geometric theory extensions (also called ‘quotients’) of the theory. Many concepts of elementary topos theory which apply to the lattice of subtoposes of a given topos are then transferred via this duality into the context of geometric theories. A second very general implementation of the ‘bridge’ technique is the investigation of the class of theories of presheaf type (i.e. classified by a presheaf topos). After establishing a number of preliminary results on flat functors in relation to classifying toposes, the book carries out a systematic investigation of this class resulting in a number of general results and a characterization theorem allowing one to test whether a given theory is of presheaf type by considering its models in arbitrary Grothendieck toposes. Expansions of geometric theories and faithful interpretations of theories of presheaf type are also investigated. As geometric theories can always be written (in many ways) as quotients of presheaf type theories, the study of quotients of a given theory of presheaf type is undertaken. Lastly, the book presents a number of applications in different fields of mathematics of the theory it develops.