Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...

. We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a set-theoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: showing that CC + Classical logic is consistent (by constructing a model for it) and showing that the Axiom of Choice is not derivable in CC (by constructing a model in which the type that represents the Axiom of Choice is empty). 1 Introduction In the literature there are many investigations on the semantics of polymorphic -calculus with dependent types (see for example [12, 11, 10, 1, 5, 13]). Most of the existing models present a semantics for systems in which the inhabitants of the impredicative universe (types) are "lifted" to inhabitants of the predicative universe (kinds) (see [16]). Such systems are convenient to be modeled by locally Cartesian-closed categories having small Cartesia...