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nLab > Latest Changes: model structure on enriched categories

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeSep 24th 2012

    Created model structure on enriched categories. I’m surprised we didn’t already have this (unless it’s under some other name I didn’t find).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2012
    • (edited Sep 24th 2012)

    Yes, true, I guess we didn’t have this yet. Thanks for creating it.

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeSep 15th 2020

    Does the model structure on CatCat-enriched categories induced by enrichment in the canonical model structure on CatCat give the correct (∞,1)(\infty, 1)-category of (2,2)(2,2)-categories? (or maybe the correct (∞,2)(\infty,2)-category or (∞,3)(\infty,3)-category of (2,2)(2,2)-categories? I’m getting mixed messages on what precisely is supposed to be presented by enrichment in model categories)

    I’ve been flipping through the various nLab pages, and I can’t find a clear statement, and this is further confounded by what seems to be inconsistent usage of 2Cat2Cat.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 15th 2020

    Added a general existence result due to Muro.

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 15th 2020
    • (edited Sep 15th 2020)

    Re #3: Have you seen https://arxiv.org/abs/1312.3881? As far as I remember, it answers your question.

    • CommentRowNumber6.
    • CommentAuthorRichard Williamson
    • CommentTimeSep 15th 2020
    • (edited Sep 15th 2020)

    The following statement on the page would imply that it gives the correct (∞,1)(\infty,1)-category of (2,2)(2,2)-categories.

    The canonical (Lack) model structure on 2Cat is induced from the canonical model structure on Cat.

    Since everything is fibrant, this amounts to checking that the weak equivalences are correct, which does seem to be the case, i.e. the description of the weak equivalences at model structure on enriched categories in this case seems to be the following one at equivalence of 2-categories:

    A 2-functor can be made into part of an equivalence iff it is essentially surjective on objects, essentially full on 1-cells (i.e. essentially surjective on Hom-categories), and fully faithful on 2-cells.

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeSep 16th 2020

    One thing that confused me was that canonical model structure on 2-categories says the weak equivalences in Lack’s model structure are the ones that also have a strict inverse (up to strict natural isomorphism). But now I see Lack’s paper actually describes what you do.

    But the other thing that was bothering me is about having enough functors. I’d been trying to find a statement a coherence theorem that said any functor between 2-categories can be transported along equivalences to a strict functor between strict 2-categories, but the only statements I could find were either just about the objects, or only made a statement involving strict 2-categories and pseudofunctors (and natural transformations and modifications).

    • CommentRowNumber8.
    • CommentAuthorAlexanderCampbell
    • CommentTimeSep 16th 2020
    • (edited Sep 16th 2020)

    Re #7: The description on the nlab page to which you link of the weak equivalences in Lack’s model structure on 2-Cat as the “strict equivalences” is incorrect.

    I’m not quite sure what you’re asking in your second paragraph. One relevant fact is that for any cofibrant 2-category AA, any 2-category BB, and any pseudofunctor F:A→BF \colon A \to B, there exists a strict 2-functor G:A→BG \colon A \to B and an invertible icon F≅GF \cong G. But probably more what you are after is the fact that for any pseudofunctor between bicategories F:A→BF \colon A \to B, there exists a strict 2-functor between strict 2-categories G:C→DG \colon C \to D and bijective-on-objects biequivalence pseudofunctors U:A→CU \colon A \to C and V:B→DV \colon B \to D such that VF=GUV F = G U; see e.g. §2.3.3 of Nick Gurski’s book on Coherence in three-dimensional category theory.

  9. Re #7: The description on the nlab page to which you link of the weak equivalences in Lack’s model structure on 2-Cat as the “strict equivalences” is incorrect

    I think that what was intended on that page was to refer to the fact that the functors are strict. I’ll correct it.

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 30th 2021

    Redirect: Dwyer-Kan model structure on enriched categories.

    diff, v4, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2023

    added publication data to this item:

    diff, v5, current

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 13th 2023

    Added a link to model structures on internal categories.

    Added:

    • Julia E. Bergner, A model category structure on the category of simplicial categories, Transactions of the American Mathematical Society 359:5 (2006), 2043-2058. arXiv, doi.

    diff, v6, current

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