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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?
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).
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?
This theorem is all about particular models, I don’t think there is any part of it that makes sense model-independently.
I made the least invasive edit I could think of that I think makes the statement true and all notations defined.
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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