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nLab > Latest Changes: Thomason model structure

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2010

    Idea-section and one further reference at Thomason model structure.

    I remember Mike once said on the blog somewhere that there might be some problem with Thomason's original claim that cofibrant objects in this structure are posets. I made a brief remark on this, but I can't find Mike's original comment.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2010
    • (edited Jan 11th 2010)

    ...

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2010

    created cohomology of a category just for completeness in the course of Mike and my discussion on the blog

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJan 12th 2010

    I can't tell you much more... all I remember is what you said (someone said there might be a problem with it). I don't even remember who it was. (-: So it could just be a malicious rumor. I certainly haven't read the proof myself.

    • CommentRowNumber5.
    • CommentAuthoradeelkh
    • CommentTimeAug 19th 2014

    I added the reference

    • CommentRowNumber6.
    • CommentAuthorZhen Lin
    • CommentTimeAug 19th 2014

    The result on calculi of left fractions is especially interesting: it’s not often that we get a privileged direction in category theory!

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 19th 2014

    Of course, the asymmetry comes from ExEx, not from category theory.

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTimeOct 10th 2019

    I pointed to the canonical model structure, for comparison.

    diff, v20, current

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeOct 10th 2019

    I undid that change, which had been unnecessary.

    diff, v20, current

    • CommentRowNumber10.
    • CommentAuthorMarc
    • CommentTimeOct 10th 2019

    updated links to Meier/Ozornova article and talk

    diff, v21, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2019

    gave these two reference more proper publication data:

    diff, v22, current

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeFeb 26th 2021

    Changed the language. The previous version gives the impression that the Thomason equivalences bear no relation to categorical equivalences, when in truth they include the categorical equivalences.

    diff, v26, current

    • CommentRowNumber13.
    • CommentAuthorHurkyl
    • CommentTimeFeb 26th 2021

    Collected some facts about the Thomason model structure

    diff, v26, current

    • CommentRowNumber14.
    • CommentAuthorHurkyl
    • CommentTimeAug 22nd 2022

    Added a formula for homotopy limits.

    diff, v30, current

    • CommentRowNumber15.
    • CommentAuthorHurkyl
    • CommentTimeAug 22nd 2022
    • (edited Aug 22nd 2022)

    (double post)

    • CommentRowNumber16.
    • CommentAuthorHurkyl
    • CommentTimeAug 22nd 2022
    • (edited Aug 22nd 2022)

    Reverted the previous edit. I think there’s an error in Hirschhorn, Model Categories and Their Localizations.

    In definition 18.1.8, the hypotheses are:

    Let M be a simplicial model category and let C be a small category. If X is a C-diagram in M

    and then proceeds to define holim(X) via the equalizer defining the end

    holim(X)=∫ c∈CX c B(C↓c) holim(X) = \int_{c \in C} X_c^{B(C \downarrow c)}

    I’m not sure I believe that statement. E.g. if we take M=sSet, let A=Δ 0A = \Delta^0 and let BB be the graph 0→1→2→3→00 \to 1 \to 2 \to 3 \to 0. Let XX be the diagram of two parallel arrows A→BA \to B, both sending the point to 00.

    BB is the 1-sphere and this equalizer should compute the loop space at 00, so the homotopy limit should be homotopy equivalent to ℤ\mathbb{Z}.

    However, this end reduces to computing the pullback of A→B×B←B Λ 2 2A \to B \times B \leftarrow B^{\Lambda^2_2}. In particular, it is a finite simplicial set, so it can’t have the right homotopy type.

    Either I’ve made an error along the way, or it’s missing a fibrancy assumption on XX.

    As an aside, the homotopy limit page has a lot more to say about colimits than limits

    diff, v33, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 22nd 2022

    Either I’ve made an error along the way, or it’s missing a fibrancy assumption on X.

    The fibrancy assumption on X is present in the correct place, see, for example, Theorem 18.5.3 in Hirschhorn.

    Hirschhorn’s homotopy limit functor must be derived in order to compute the correct answer.

    This is an unfortunate mismatch with older terminology (including Bousfield–Kan), who use “homotopy (co)limit” to refer to what we would call “weighted (co)limit” (with respect to a certain choice of weight).

    These weighted (co)limits do compute homotopy (co)limits, but only under additional (co)fibrancy conditions.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2023

    added pointer to:

    diff, v35, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2023

    added pointer to:

    diff, v35, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2023

    added pointer to

    (though have any of Bruckner’s articles been published?)

    diff, v35, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2023
    • (edited May 1st 2023)

    have fixed a few oddities in the formulation of the definition (here)

    also deleted this line from the References-section:

    [In Cisinski 1999] it was clarified that every cofibrant object in the Thomason model structure is a poset (although this is already explicitly mentioned in Thomason’s paper – see the beginning of section 5).

    and instead added explicit statement of and reference to Thomason’s proposition 5.7 here

    diff, v35, current

1 to 21 of 21

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