Univalent categories and the Rezk completion

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of " category " for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type of isomorphisms 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 universal 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. Of the branches of mathematics, category theory is one which perhaps fits the least comfortably into existing " foundations of mathematics ". This is true both at an informal level, and when trying to be completely formal using a computer proof assistant. One problem is that naive category theory tends to run afoul of Russellian paradoxes and has to be reinterpreted using universe levels; we will not have much to say about this. But another problem is that most of category theory is invariant under weaker notions of " sameness " than equality, such as isomorphism in a category or equivalence of categories, in a way which traditional foundations (such as set theory) fail to capture. This problem becomes especially important when formalizing category theory in a computer proof assistant. Our aim in this paper is to show that this problem can be ameliorated using the new Univalent Foundations of mathematics, a.k.a. homotopy type theory, proposed by Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2 Voevodsky (2010). It builds on the existing system of dependent type theory (Martin-Löf 1984; Werner 1994), a logical system that is feasible for large-scale formalization of mathematics (Gonthier et al. 2012) and also for internal categorical logic. The distinctive feature of Univalent Foundations (UF) is its treatment of equality inspired by homotopy-van den Berg and Garner 2012). Using this interpretation, Voevodsky has extended dependent type theory with an additional axiom, called the Univalence Axiom, which was originally suggested by the model of the theory in the category of …