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.
Original language | English |
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Pages (from-to) | 1010-1039 |
Number of pages | 30 |
Journal | Mathematical Structures in Computer Science |
Volume | 25 |
Issue number | 5 |
Early online date | 19 Jan 2015 |
DOIs | |
Publication status | Published - 1 Jun 2015 |