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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 17th 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 18th 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 31st 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.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 11th 2022

    Added:

    There are analogues of the Joyal model structure for cubical sets (with or without connection):

    diff, v25, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 30th 2022

    Added:

    History

    André Joyal on the history of the Joyal model structure (also on MathOverflow):

    I became interested in quasi-categories (without the name) around 1980 after attending a talk by Jon Beck on the work of Boardman and Vogt. I wondered if category theory could be extended to quasi-categories. In my mind, a crucial test was to show that a quasi-category is a Kan complex if its homotopy category is a groupoid. All my attempts at showing this have failed for about 15 years, until I stopped trying hard! I found a proof after extending to quasi-categories a few basic notions of category theory. This was around 1995. The model structure for quasi-categories was discovered soon after. I did not publish it immediately because I wanted to show that it could be used for proving something new in homotopy theory. I am a bit of a perfectionist (and overly ambitious?). I was hoping to develop a synthesis between category theory and homotopy theory (hence the name quasi-categories). I met Lurie at a conference organised by Carlos Simpson in Nice (in 2001?). I gave a talk on the model structure and Lurie asked for a copy of my notes afterward.

    diff, v26, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023
    • (edited May 29th 2023)

    expanded out the line (in the Definition):

    …the left adjoint of the homotopy coherent nerve to a weak equivalence in the model structure on simplicial categories

    to

    …the rigidification functor ℭ\mathfrak{C} (the left adjoint to the homotopy coherent nerve) to a Dwyer-Kan equivalence (a weak equivalence in the Dwyer-Kan-Bergner model structure on sSet-enriched categories).

    diff, v29, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2023

    have adjusted the typesetting of the Bousfield localization adjunction (here)

    sSet Quillen⊥⟶Id⟵IdsSet Joyal sSet_{Quillen} \underoverset {\underset{Id}{\longrightarrow}} {\overset{Id}{\longleftarrow}} {\;\; \bot \;\;} sSet_{Joyal}

    diff, v31, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2023

    added publication data for

    and fixed the broken links to this item

    diff, v31, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2023

    added the statement (here) that Cartesian product preserves weak categorical equivalences

    diff, v32, current

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