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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum 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 sheaves 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.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 28th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea Right Bousfield localizations of [[model categories]] are a way to encode [[coreflective localizations]] of [[(∞,1)-categories]] in the setting of [[model categories]]. Given a [[model category]] $M$, the formal definition and elementary properties can be deduced from the definition and properties of [[left Bousfield localizations]] of the [[opposite category]] of $M$. However, the typical existence theorems for right Bousfield localizations are not formulated in this manner, since typical conditions involve some form of [[local presentability]] of the underlying category, or an analogous condition, which typically do not hold for the [[opposite category]]. ## Barwick's existence theorem If $C$ is a [[tractable model category]], $D$ is an accessible accessibly embedded subcategory of $C$ that is closed under [[homotopy colimits]] in $C$. Then there is a right Bousfield localization of $C$, which is a [[right model category]] (but not necessarily a [[model category]]), and its colocal objects are precisely the object of $D$. ## Related concepts * [[left Bousfield localization of model categories]] ## References The original article: * {#Bousfield75} [[A. K. Bousfield]], *The localization of spaces with respect to homology*, Topology **14** (1975), 133–150 &lbrack;<a href="http://dx.doi.org/10.1016/0040-9383(75)90023-3">doi:10.1016/0040-9383(75)90023-3</a>&rbrack; Detailed discussion (including existence results for [[left proper model category|left proper]] [[cellular model category|cellular model categories]]): * {#Hirschhorn02} [[Philip Hirschhorn]], Chapter 5 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs 99 (2002) &lbrack;[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)&rbrack; Existence results for [[combinatorial model categories]]: * {#Barwick10} [[Clark Barwick]], *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320 &lbrack;[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)&rbrack; Right Bousfield localization specifically for [[stable model categories]] (such as [[model structures on spectra]]): * D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations. Glasg. Math. J., 56(1):13–42, 2014. [[!redirects right Bousfield localization]] <a href="https://ncatlab.org/nlab/revision/right+Bousfield+localization+of+model+categories/1">v1</a>, <a href="https://ncatlab.org/nlab/show/right+Bousfield+localization+of+model+categories">current</a>

   Created:

   Idea

   Right Bousfield localizations of model categories are a way to encode coreflective localizations of (∞,1)-categories in the setting of model categories.

   Given a model category $M$, the formal definition and elementary properties can be deduced from the definition and properties of left Bousfield localizations of the opposite category of $M$.

   However, the typical existence theorems for right Bousfield localizations are not formulated in this manner, since typical conditions involve some form of local presentability of the underlying category, or an analogous condition, which typically do not hold for the opposite category.

   Barwick’s existence theorem

   If $C$ is a tractable model category, $D$ is an accessible accessibly embedded subcategory of $C$ that is closed under homotopy colimits in $C$. Then there is a right Bousfield localization of $C$, which is a right model category (but not necessarily a model category), and its colocal objects are precisely the object of $D$.

   Related concepts

   • left Bousfield localization of model categories

   References

   The original article:

   • {#Bousfield75} A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150 [doi:10.1016/0040-9383(75)90023-3]

   Detailed discussion (including existence results for left proper cellular model categories):

   • {#Hirschhorn02} Philip Hirschhorn, Chapter 5 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf]

   Existence results for combinatorial model categories:

   • {#Barwick10} Clark Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 2 (2010) 245–320 [doi:10.4310/hha.2010.v12.n2.a9, euclid:1296223884, subsuming: arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]

   Right Bousfield localization specifically for stable model categories (such as model structures on spectra):

   • D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations. Glasg. Math. J., 56(1):13–42, 2014.

   !redirects right Bousfield localization

   v1, current