Doctrines in Categorical Logic

Abstract

Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on model theory and set theory. It is organized by increasing richness of the doctrines involved. These doctrines are categorical analogs of fragments of logical theories that have sufficient category-theoretic structure for their models to be described as functors. The introduction of the categorical notion of algebraic theory led to a systematic theory of relative interpretations of one equational theory into another, as well as a theory about the categories (or varieties) of algebras for these, and their relationship. This progress springs from having a presentation-invariant notion of equational (or algebraic) theory. The chapter also deals with doctrines of equational, Cartesian, finitary coherent, and infinitary coherent logic. Higher order logic and set theory are also discussed in the chapter.