Not signed in (Sign In)

Start a new discussion

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: Bousfield localization of model categories

Bottom of Page

1 to 6 of 6

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2009
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2009

    I completed the existece proof with the bit showing the accessibility of the S-local weak equivalences.

    Then I added a section prerequisite for the proof that lists briefly all the material at other entries, previously just having been linked to, that enters the proof.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2009
    • (edited Nov 26th 2009)

    Based on some feedback that I got behind the scenes, I have now rearranged the material, to make it more light-weight at the beginning.

    • the section Properties now has the central properties one can deduce for a Bousfield localization that do not require combinatorial model category technology

    • the section Existence of ... has the existence result for the combinatorial case pretty much as before, but minus the material that has been moved to "Properties"

    • and the Definition section is now such that it downplays the "derived hom" \mathbf{R}Hom(-,-) and instead tends to arrange things such that they are properly cofibrant or fibrant . Because it was suggested to me that invoking the derived hom concept makes this look, to the non-expert reader- much more arcane than it is.

    I just hope that the interconnecting logic between the paragraphs is now still consistent. I fddled a bit with adjusting the statements to the slightly changed assimptions.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2012

    added the statement that left Bousfield localization is functorial with respect to Quillen equivalences, here.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2017
    • (edited Mar 17th 2017)

    I have added statement and proof (here) that given a left Bousfield localization, then local fibrations between local objects are global fibrations.

    (I am assuming functorial factorization, which is of course provided under the generic assumption of the entry that we are localizing a cofibrantly generated model category. But probably one can get the statement more generally.)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2017
    • (edited Mar 17th 2017)

    I have made more explicit statement and proof that functorial fibrant replacement in a left Bousfield localized model category is a model for the derived adjunction unit, here

1 to 6 of 6

  1.  
Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.

    • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)
 
Back to Discussions Top of Page