Sets, Topoi, and Internal Logic in Categories

Abstract

Publisher Summary A recent development in category theory has been the description and study of a useful class of categories called “elementary topoi.” This class includes the ordinary category of sets: diagrams of sets and sheaves of sets on a topological space. The axiomatic description of these categories provides a formulation of axiomatic set theory wholly different from the usual set-theoretic axioms on the membership relation and the further study casts considerable light on a number of problems of foundations. Systematic developments of the properties of a topos from axioms are discussed in this chapter. Other categories with internal logic and quantifiers and their relations to standard logical systems are also described in the chapter. Given the basic geometric character of sheaf theory, this common development of ideas from geometry and concepts from set theory embodies exciting prospects.