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1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeJun 9th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexcopying reference from HoTT wiki Anonymous <a href="https://ncatlab.org/nlab/revision/Univalent+categories+and+the+Rezk+completion/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Univalent+categories+and+the+Rezk+completion">current</a>

   copying reference from HoTT wiki

   Anonymous

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have hyperlinked the abstract, and I have taken the liberty of reformatting as follows -- what do you think: *** * [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]]: **Univalent categories and the Rezk completion** Math. Structures Comput. Sci. **25** 5 (2015) 1010-1039. [arXiv:1303.0584](http://arxiv.org/abs/1303.0584) [doi:10.1017/S0960129514000486](https://doi.org/10.1017/S0960129514000486) on [[internal categories in homotopy type theory]] and their [[Rezk completion]] to [[univalent categories]]. > **Abstract.** We develop [[category theory]] within [[Univalent Foundations]], which is a [[mathematical foundation|foundational system for mathematics]] based on a [[homotopy type theory|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|Univalence Axiom]], saying that the [[type]] of [[isomorphisms]] between any two [[objects]] is [[equivalence in homotopy type theory|equivalent]] to the [[identity type]] between these objects; we call them "saturated" or [[univalent categories|"univalent" categories]]. Moreover, we show that any category is [[weak equivalence|weakly equivalent]] to a univalent one in a [[universal construction|universal]] [[Rezk completion|way]]. In [[relation between category theory and type theory|homotopical and higher-categorical semantics]], this construction corresponds to a truncated version of the [[Rezk completion]] for [[Segal spaces]], and also to the [[stackification|stack completion]] of a [[prestack]]. *** <a href="https://ncatlab.org/nlab/revision/diff/Univalent+categories+and+the+Rezk+completion/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Univalent+categories+and+the+Rezk+completion/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Univalent+categories+and+the+Rezk+completion">current</a>

   I have hyperlinked the abstract, and I have taken the liberty of reformatting as follows – what do you think:

   ---

   • Benedikt Ahrens, Chris Kapulkin, Michael Shulman:

     Univalent categories and the Rezk completion

     Math. Structures Comput. Sci. 25 5 (2015) 1010-1039.

     arXiv:1303.0584

     doi:10.1017/S0960129514000486

   on internal categories in homotopy type theory and their Rezk completion to univalent categories.

   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.

   ---

   diff, v2, current