Univalent categories and Rezk completion

is a foundational system for mathematics based on a homotopical interpre-tation of dependent type theory. In this system, we propose a definition of “category ” for which equality and equivalence of categories agree. Such cate-gories satisfy a version of the Univalence Axiom, saying that the type of iso-morphisms between any two objects is equivalent to the identity type between these objects; we call them “saturated ” or “univalent ” categories. Moreover, we show that any category is weakly equivalent to a univalent one in a uni-versal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack