Abstract In this paper, we define indexed type theories which are related to indexed (∞-)categories in the same way as (homotopy) type theories are related to (∞-)categories. We define several standard constructions for such theories including finite (co)limits, arbitrary (co)products, exponents, object classifiers, and orthogonal factorization systems. We also prove that these constructions are equivalent to their type theoretic counterparts such as Σ-types, unit types, identity types, finite higher inductive types, Π-types, univalent universes, and higher modalities.

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-Mod of its models carries the structure of a model category. We also show that if T has Σ-types, then weak equivalences can be characterized in terms of homotopy categories of models.