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    • CommentRowNumber1.
    • CommentAuthorBaptiste
    • CommentTimeOct 6th 2017


    In the (infinity,1)-functor page, in part Properties, there is a Theorem. Some notation is introduced in it, [C^op,KanCplx]°, but instead [C^op,sSet]° is used in the statement. Is the statement incorrect?

    Also, I guess it would be nice to add a reference for this Theorem. Does anyone have one?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 6th 2017

    Yes, this theorem statement is a bit confused. The fibrant and cofibrant objects are all valued in Kan complexes, but a consistent notation should be used. Also the text says we have an equivalence of \infty-groupoids, but the displayed equation is an equivalence between (,1)(\infty,1)-categories.

    For a reference, if you follow enough links you can find a citation to Lurie at (infinity,1)-category of (infinity,1)-functors (models).

    • CommentRowNumber3.
    • CommentAuthorBaptiste
    • CommentTimeOct 9th 2017

    Thank you.

    I understand the statement in the reference to Lurie.

    I’m not making the edit to clarify the entry under discussion, because I’m not sure how to do it. I don’t think it’s a good idea, in a statement, to mix concrete models for (,1)(\infty,1)-categories (such as quasicategories, at the beginning of the statement) and then general (,1)(\infty,1)-notions that should in principle existe for any model (such as (,1)(\infty,1)-functor later on). Here the final equivalence in the statement seems to live in the world of simplicial categories. So what should be done?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeOct 9th 2017

    This theorem is all about particular models, I don’t think there is any part of it that makes sense model-independently.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 16th 2017

    I made the least invasive edit I could think of that I think makes the statement true and all notations defined.

    • CommentRowNumber6.
    • CommentAuthorBaptiste
    • CommentTimeOct 23rd 2017

    Thanks. It looks better indeed.

    I understand that this theorem is about particular models; actually, it seems to be about the particular model of quasi-categories. That is why I believe the statement should end by

    Then we have an equivalence of quasi-categories etc. But it is a minor point, I guess.

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