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nLab > Latest Changes: model structure for quasi-categories

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2013
    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 16th 2018

    Added an explicit description of generating acyclic cofibrations for the Joyal model structure.

    diff, v16, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2018

    Cleaned up statement of Stevenson’s theorem about the generating acyclic cofibrations and added the arXiv reference to his paper. (I talked with Danny about this yesterday and he agrees that the hypothesis can be stated as the equivalent ’countably-many simplices’, rather than ’countably-many non-degenerate simplices’)

    Richard, if you are reading this, the LaTeX-like syntax that Dmitri used isn’t working. I left it in so this could be verified and fixed, if indeed there is abug.

    diff, v17, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 17th 2018

    An element of B is a simplicial set, not a map of simplicial sets. What does it mean for a simplicial set to be an acyclic cofibration?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2018
    • (edited Oct 17th 2018)

    Gah, there’s a bijection ℬ→{{0}→B∣B∈ℬ}\mathcal{B}\to \{\{0\}\to B\mid B\in \mathcal{B}\}, and I was implicitly casting along this.

  6. Richard, if you are reading this, the LaTeX-like syntax that Dmitri used isn’t working. I left it in so this could be verified and fixed, if indeed there is a bug.

    Thanks for the message. I asked Dmitri to leave this; it is not functionality that has been implemented yet, but I have been planning to add it very soon. But feel free to change it to the more conventional syntax in the meantime, I think I will remember that the reference was added here.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 17th 2018

    Fixed the issue that Dmitri pointed out.

    diff, v18, current

  8. The Joyal model structure has, in fact, a simplicial enrichment compatible with the model structure (although of course not the canonical one). I’ll add the details later but I wanted to remove the wrong statement

    Denis Nardin

    diff, v19, current

  9. I removed the incorrect (as I explained here: https://chat.stackexchange.com/transcript/message/49892813#49892813) generating set of trivial cofibrations.

    I also corrected wording in a few places and added references for Joyal’s original construction of the model structure.

    Alexander Campbell

    diff, v21, current

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeJan 30th 2021
    • (edited Jan 30th 2021)

    Is comment #8 correct, in that the Joyal model structure is simplicial with regard to another enrichment? Is the idea using Map(X,Y) n=Hom(Δ′[n]×X,Y)Map(X,Y)_n = Hom(\Delta'[n] \times X, Y), or do you need something more complicated?

    • CommentRowNumber11.
    • CommentAuthorRichard Williamson
    • CommentTimeJan 30th 2021
    • (edited Jan 30th 2021)

    I’m not sure what Denis had in mind, but intuitively one should be able to enrich over the Quillen model structure by replacing the internal homs of the Joyal model structure by their ’cores’ (i.e. throw away non-invertible 1-arrows).

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeJan 30th 2021
    • (edited Jan 30th 2021)

    Enriching qCatqCat like that (which I will call MapMap) does give a useful simplicially enriched category (which Lurie uses in his definition of Cat ∞Cat_\infty) but that enrichment doesn’t correspond in an immediately obvious way to a Quillen-simpicially enriched category structure on sSetsSet.

    (Lurie’s exposition implies that the reason for considering marked simplicial sets is precisely to realize that enrichment of qCatqCat as the fibrant-cofibrant subcategory of a simplicial model structure on sSet +sSet^+)

    • CommentRowNumber13.
    • CommentAuthorRichard Williamson
    • CommentTimeJan 30th 2021
    • (edited Jan 30th 2021)

    Ah, I see the kind of issue that can arise now. It is only reasonable for Core(Y X)Core(Y^X) itself to be appropriate when both XX and YY are quasi-categories. Perhaps one can tweak things in the case that XX is not a quasi-category? Maybe XX should be replaced by a Kan complex, for example using the Ex ∞Ex^\infty functor, which has good preservation properties? Of course one has to check that this even makes sense, i.e. gives an enrichment of the original Hom set.

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