Univalent categories and the Rezk completion

Benedikt Ahrens, Krzysztof Kapulkin, Michael Shulman

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1010-1039
Number of pages30
JournalMathematical Structures in Computer Science
Issue number5
Early online date19 Jan 2015
Publication statusPublished - 1 Jun 2015


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