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nLab > Latest Changes: model structure on simplicial presheaves

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2010

    added more details on weak equivalences (local epimorphisms, really) in the Cech localization of the projective model structure from Dugger-Hollander-Isaksen

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 24th 2011
    • (edited Jan 24th 2011)

    at model structure on simplicial presheaves I have (finally) added a section

    Presentation of (oo,1)-toposes

    I have also added to the section Homtopy (co)limits the observation that finite homotopy limits in the local model structure may be computed in the global structure.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 24th 2011

    have added also a section Inclusion of chain complexes of sheaves, so far just observing the obvious Quillen adjunction induced from Dold-Kan

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 1st 2011
    • (edited Feb 1st 2011)

    I had had a section on descent for presheaves with coefficients in strict ∞\infty-groupoids and how it relates to the descent of these regarded as simplicial presheaves over in the entry on smooth ∞\infty-groupoids. Since it did not really belog there specifically at all, I have now moved that over to model structure on simplicial presheaves in a new section:

    Descent for values in strict and abelian ∞\infty-groupoids.

    The main point is there a clear statement of Verity’s result of sufficient conditions under which Street’s definition of descent is actually correct and matches the one of simplicial presheaves.

    (I have only copy-and-pasted it for the moment. I should go through this and see if there is need to polish or otherwise improve this.)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2018

    added pointer to the MO comment Garner 13 characterizing projectively cofibrant simplicial presheaves.

    Somebody should spell out the details here! If only by copy-and-pasting-and-beautifying-a-little the code from the MO answer.

    diff, v130, current

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeMar 29th 2019

    Fixed some ?ech which should have been Čech.

    diff, v134, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2020

    I have added publication data to

    diff, v137, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2021
    • (edited Jun 12th 2021)

    cross-linked the discussion of local right properness (here) with locally cartesian closed model category and with locally cartesian closed (infinity,1)-category

    diff, v142, current

    • CommentRowNumber10.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 12th 2022

    Proposition 6.3 at model structure on simplicial presheaves claims that the localization of presheaves of spaces at a coverage is a topological localization, but I do not understand the proof. The second commuting square is claimed to be a pushout in which the top morphism is a monomorphism, and I do not see why either claim holds. It further seems to me that the sieve S({U i}∪{V j,k})S(\{U_i\}\cup \{V_{j,k}\}) is just the same as S({U i})S(\{U_i\}) since each V j,kV_{j,k} refines some U iU_i by definition, so the conclusion of the proof also does not make sense to me.

    This is related to a recent answer I wrote on MathOverflow, where I explain the proof of a weaker claim (for pullback-stable coverages).

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2022

    Thanks for the heads-up. Looks like I wrote this in May 2010 (where I had still a disclaimer which I seem to have later removed), but it is indeed quite mangled. I now went ahead and fixed up that Prop. 6.3, but looking ahead this is not the only edit necessary.

    So for the moment I have moved that whole chunk (former Prop. 6.3 up to the Conclusion 6.6) to the Sandbox. Will see what to do with it….

    • CommentRowNumber12.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 12th 2022

    Looking at what’s in the sandbox now, I think that for the square in the proof of Prop 0.1 to be a pushout, one should replace y(V)y(V) by the sieve generated by VV. Otherwise there is no reason for the pushout to be a subobject of y(U)y(U).

    I was also looking at the proof in Johnstone (C, Lemma 2.1.6), which is basically this: If R⊂S⊂y(X)R\subset S\subset y(X) are sieves such that RR is in the (sifted) coverage and if FF is a sheaf for the coverage, then Map(S,F)→Map(R,F)Map(S,F)\to Map(R,F) has a section, in particular is an effective epimorphism. On the other hand it is the limit of the maps F(V)→Map(R× y(X)y(V),F)F(V) \to Map(R\times_{y(X)}y(V),F) for VV in the sieve SS, which by definition of coverage have retractions since each R× y(X)y(V)R\times_{y(X)}y(V) contains a covering sieve. For a sheaf of sets this means that these maps are monomorphisms and we can conclude. But already for a sheaf of groupoids I’m not sure what to do…

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