_Univalent categories and the Rezk completion_. [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], Math. Structures Comput. Sci. 25 (2015), no. 5, 1010?1039. ## Links ## * [arXiv:1303.0584](http://arxiv.org/abs/1303.0584) * [[nLab:internal categories in HoTT]] ### Abstract ### 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. category: reference