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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory 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 limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural 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 string 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).
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 31st 2021

    Redirect: Barwick-Kan equivalence.

    Renamed.

    diff, v7, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeFeb 26th 2021

    The fact the Quillen equivalence is actually an adjoint weak equivalence of relative categories is significant enough that I reorganized things a bit so I can state it.

    It also seems useful to put a little more emphasis on the fact that RelCat can model simplicial spaces.

    diff, v9, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeFeb 27th 2021

    Added the compatibility with simplicial localization.

    diff, v10, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeFeb 27th 2021

    Added the connection to marked simplicial sets, as well as the description of (C,W) as modeling the localization C[W 1]C[W^{-1}].

    diff, v11, current

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeFeb 27th 2021

    I seem to have broken the formatting system in the statement of the theorem that RelCat(,1)CatRelCat \to (\infty,1)Cat is (C,W)C[W 1](C,W) \to C[W^{-1}], and I can’t figure out what I’ve done wrong.

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeFeb 27th 2021

    By the way, does nLab have an established notation for localization of \infty-categories? I’m uncomfortable with C[W 1]C[W^{-1}] due to the risk of that being interpreted as the localization of 1-categories.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 27th 2021

    Fixed formatting.

    diff, v12, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2021
    • (edited Feb 27th 2021)

    @Hurkyl #6,

    I like to use “L WL_W”, following/alluding to standard (?) notation for Dwyer-Kan simplicial localizations, such as Hammock localization “L W HL^H_W”.

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeFeb 28th 2021
    • (edited Feb 28th 2021)

    L W(C)L_W(C) or L(C,W)L(C,W)? I guess the difference is whether you are in a context viewing CC or (C,W)(C,W) as the primary object of interest. I’ve switched over the notation.

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)