Dependent Cartesian Closed Categories

We present a generalization of cartesian closed categories (CCCs) for dependent types, called dependent cartesian closed categories (DCCCs), which also provides a reformulation of categories with families (CwFs), an abstract semantics for Martin-Löf type theory (MLTT) which is very close to the syntax. Thus, DCCCs accomplish mathematical elegance as well as a direct interpretation of the syntax. Moreover, they capture the categorical counterpart of the generalization of the simply-typed λ-calculus (STLC) to MLTT in syntax, and give a systematic perspective on the relation between categorical semantics for these type theories. Furthermore, we construct a term model from the syntax, establishing the completeness of our interpretation of MLTT in DCCCs.