A Note on Generalized Algebraic Theories and Categories with Families

Abstract

We give a new syntax independent definition of the notion of a generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature $\Sigma$ for a generalized algebraic theory and the associated category of cwfs with a $\Sigma$-structure and cwf-morphisms that preserve this structure on the nose. Our definition refers to uniform families of contexts, types, and terms, a purely semantic notion. Furthermore, we show how to syntactically construct initial cwfs with $\Sigma$-structures. This result can be viewed as a generalization of Birkhoff's completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer's construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of dependent type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a category with families.