Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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