Model Structures on Categories of Models of Type Theories

Abstract

Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory $T$ has enough structure, then the category $T\text{-}\mathbf{Mod}$ of its models carries the structure of a model category. We also show that if $T$ has $\Sigma$ types, then weak equivalences can be characterized in terms of homotopy categories of models.