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• CommentRowNumber1.
• CommentAuthorBaptiste
• CommentTimeOct 6th 2017

Hello,

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 $(\infty,1)$-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 $(\infty,1)$-categories (such as quasicategories, at the beginning of the statement) and then general $(\infty,1)$-notions that should in principle existe for any model (such as $(\infty,1)$-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.