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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2010

    a few more details at model structure for left fibrations

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeDec 18th 2011

    The page model structure for left fibrations claims that this model structure is proper. However, the cited propositions in HTT only claim that it is left proper. Is it right proper? If so, where is a proof?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2011

    Thanks for catching this. I have fixed it. Not sure why the “left” got missing.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 30th 2015

    Suppose XX is a quasicategory. The model structure for left fibrations on sSet/XsSet/X is a left Bousfield localization of the model structure on an over category arising from the model structure for quasi-categories, since it has the same cofibrations and fewer fibrant objects. Can one exhibit a nice small set of maps at which it is the localization?

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeJul 1st 2015
    • (edited Jul 1st 2015)

    Well, the left model structure on sSet /X\mathbf{sSet}_{/ X} is enriched with respect to the Kan–Quillen model structure on sSet\mathbf{sSet}, and all objects are cofibrant, so the following is true:

    • A morphism f:ABf : A \to B in sSet /X\mathbf{sSet}_{/ X} is a weak equivalence in the left model structure if and only if f *:Hom̲(B,C)Hom̲(A,C)f^* : \underline{Hom}(B, C) \to \underline{Hom}(A, C) is a (homotopy) equivalence of Kan complexes for every left fibration CC over XX.

    (I am abusing notation by omitting the projections to XX.) On the other hand, given a set 𝒮\mathcal{S} of morphisms in sSet /X\mathbf{sSet}_{/ X}, since the slice Joyal model structure is enriched with respect to the Joyal model structure on sSet\mathbf{sSet}, the following should be true:

    • An isofibration CC over XX is an 𝒮\mathcal{S}-local object in sSet /X\mathbf{sSet}_{/ X} if and only if f *:Hom̲(B,C)Hom̲(A,C)f^* : \underline{Hom}(B, C) \to \underline{Hom}(A, C) is a (categorical) equivalence of quasicategories for every f:ABf : A \to B in 𝒮\mathcal{S}.

    Putting these two together, it would seem to me that to get the desired 𝒮\mathcal{S}, it is enough to check that the 𝒮\mathcal{S}-local objects you get are the left fibrations. So I think 𝒮={{0}Δ n:n0}\mathcal{S} = \{ \{ 0 \} \hookrightarrow \Delta^n : n \ge 0 \} works.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 1st 2015

    Hmm, when you have an enriched model structure I think you actually get two different notions of locality depending on whether you use the enriched hom-objects or the model-categorical hom-spaces, and the latter is the one that corresponds to the usual sort of Bousfield localization. So I don’t believe your second bullet; instead I would expect to see f *f^* acting on maximal sub-Kan-complexes of those mapping spaces.

    Secondly, how do you know that your 𝒮\mathcal{S} (which is the obvious choice) does in fact detect the left fibrations?

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeJul 1st 2015
    • (edited Jul 1st 2015)

    Hmmm, maybe the 𝒮\mathcal{S} I suggested doesn’t quite work. How about 𝒮={Λ 0 nΔ n:n>0}\mathcal{S} = \{ \Lambda^n_0 \hookrightarrow \Delta^n : n \gt 0 \} instead?

    1. If f:ABf : A \to B is a monomorphism and CC is an isofibration, then f *:Hom̲(B,C)Hom̲(A,C)f^* : \underline{Hom}(B, C) \to \underline{Hom}(A, C) is an isofibration.
    2. So if CC is an isofibration and 𝒮\mathcal{S}-local in the Joyal-enriched sense, then CC has the right lifting property with respect to all members of 𝒮\mathcal{S}, hence is a left fibration.
    3. If CC is a left fibration, then it is 𝒮\mathcal{S}-local in the Joyal-enriched sense, because the members of 𝒮\mathcal{S} are weak equivalences in the left model structure.
    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJul 2nd 2015

    Okay. And I think your argument works for ordinary localization too, replacing the internal hom-objects with their maximal subgroupoids and “isofibration” with Kan fibration.

    Pedro Boavida has just pointed out to me that your 𝒮\mathcal{S} in #7 is also the answer asserted by Moerdijk-Heuts on page 5 here.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 2nd 2015

    By the way, I assumed you meant the set of all maps of the form Λ 0 nΔ n\Lambda^n_0 \hookrightarrow \Delta^n over XX, i.e. indexed not just by nn but by a map Δ nX\Delta^n\to X.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJul 2nd 2015

    I have added a mention of this fact to the page model structure for left fibrations.

    • CommentRowNumber11.
    • CommentAuthorZhen Lin
    • CommentTimeJul 3rd 2015

    Re #9: Yes, of course. One of the irritations of this abuse of notation.

    The definition of weak equivalence appearing on the page (and in HTT) is rather mysterious. (It is much too clever for a definition!) I think I like this Bousfield localisation definition better.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 4th 2015

    But can you show that the fibrant objects in the Bousfield localization are precisely the left fibrations without already knowing that the model structure as defined by Lurie exists?

    • CommentRowNumber13.
    • CommentAuthorZhen Lin
    • CommentTimeJul 4th 2015
    • (edited Jul 4th 2015)

    Well, let’s see. We know every 𝒮\mathcal{S}-local object is a left fibration. Conversely, an isofibration CSC \to S is 𝒮\mathcal{S}-local in the Joyal-enriched sense if and only if it has the right lifting property with respect to Δ m×Λ 0 nΔ m×Δ nΔ m×Δ n\Delta^m \times \Lambda^n_0 \cup \partial \Delta^m \times \Delta^n \hookrightarrow \Delta^m \times \Delta^n. But that inclusion is a left anodyne extension by [HTT, Corollary 2.1.2.7], so we are done.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJul 4th 2015

    Every time I say “Bousfield localization”, I mean in the ordinary sense, not the Joyal-enriched sense.

    • CommentRowNumber15.
    • CommentAuthorZhen Lin
    • CommentTimeJul 5th 2015

    It’s obvious that 𝒮\mathcal{S}-local in the Joyal-enriched sense implies 𝒮\mathcal{S}-local in the ordinary sense (for fibrant objects).

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 5th 2015

    Okay, fair enough. Although HTT 2.1.2.7 is already more than half the way towards Lurie’s construction of the left fibration model structure.

  1. Hi, sorry to trouble you; I guess also in #13 above it is being implicitly used that every left fibration is an isofibration? (Of course this is well known to be true, probably the most direct way to see this is to use the description of covariant equivalences in [HTT, Definition 2.1.4.5]).
    • CommentRowNumber18.
    • CommentAuthorZhen Lin
    • CommentTimeJul 7th 2015

    Sure. That’s also a formal consequence of the claim that the left fibration model structure is a left Bousfield localisation of the sliced Joyal model structure.

  2. In theorem 3.3 describing a set of maps to localize at, the left horn was mistakenly notated with a \Delta rather than a \Lambda.

    Anonymous

    diff, v18, current

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