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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 31st 2021

    Redirect: Barwick-Kan equivalence.


    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.

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