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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/Localization+in+Homotopy+Type+Theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Localization+in+Homotopy+Type+Theory">current</a>

   copying reference from HoTT wiki

   Anonymous

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexJust on aesthetics: It seems a little heavy to markup an article reference as a section _Localization in Homotopy Type Theory_, [[Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], ## Links ## [arXiv:1807.04155](https://arxiv.org/abs/1807.04155) Why not use more usual bibitem formatting, such as: * [[Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], *Localization in Homotopy Type Theory*, Higher Structures **4** 1 (2020) 1-32 $[$[arXiv:1807.04155](https://arxiv.org/abs/1807.04155), [doi](https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/ChrOpiRijSco)$]$ This looks more professional, and has the advantage that one can copy-and-paste it for use as an actual reference. In fact, this is the form in which this article is being referenced at various places on the nLab, for instance [here](https://ncatlab.org/nlab/show/J.+Daniel+Christensen#CORS20). <a href="https://ncatlab.org/nlab/revision/diff/Localization+in+Homotopy+Type+Theory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Localization+in+Homotopy+Type+Theory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Localization+in+Homotopy+Type+Theory">current</a>

   Just on aesthetics:

   It seems a little heavy to markup an article reference as a section

   _Localization in Homotopy Type Theory_, [[Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], 
   
   ## Links ##
   
   [arXiv:1807.04155](https://arxiv.org/abs/1807.04155)
   

   Why not use more usual bibitem formatting, such as:

   • Daniel Christensen, Morgan Opie, Egbert Rijke, Luis Scoccola,

     Localization in Homotopy Type Theory,

     Higher Structures 4 1 (2020) 1-32

     $[$arXiv:1807.04155, doi$]$

   This looks more professional, and has the advantage that one can copy-and-paste it for use as an actual reference.

   In fact, this is the form in which this article is being referenced at various places on the nLab, for instance here.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn adding the missing publication data, I have taken the liberty of re-formatting as follows, what do you think: *** * [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]]: **Localization in Homotopy Type Theory** Higher Structures **4** 1 (2020) 1-32 [arXiv:1807.04155](https://arxiv.org/abs/1807.04155) on [[localization]] in [[homotopy type theory]]. > **Abstract.** We study [[localization]] [[p-localization|at a prime]] in [[homotopy type theory]], using [[endomorphism|self maps]] of the [[circle]]. Our main result is that for a [[pointed]], [[simply connected]] [[type]] $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all [[homotopy groups]]. In order to prove this, we further develop the theory of [[reflective subuniverses]]. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L_0$. Furthermore, we prove results establishing that $L_0$ is almost [[left exact]]. We next focus on localization with respect to a map, giving results on preservation of [[coproducts]] and [[connectivity]]. We also study how such localizations interact with other reflective subuniverses and [[orthogonal factorization systems]]. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an [[Eilenberg-Mac Lane space]] $K(G, n)$ with $G$ abelian. We also include a partial converse to the main theorem. *** <a href="https://ncatlab.org/nlab/revision/diff/Localization+in+Homotopy+Type+Theory/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Localization+in+Homotopy+Type+Theory/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Localization+in+Homotopy+Type+Theory">current</a>

   In adding the missing publication data, I have taken the liberty of re-formatting as follows, what do you think:

   ---

   • J. Daniel Christensen, Morgan Opie, Egbert Rijke, Luis Scoccola:

     Localization in Homotopy Type Theory

     Higher Structures 4 1 (2020) 1-32

     arXiv:1807.04155

   on localization in homotopy type theory.

   Abstract. We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L_0$. Furthermore, we prove results establishing that $L_0$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space $K(G, n)$ with $G$ abelian. We also include a partial converse to the main theorem.

   ---

   diff, v4, current