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Somebody kindly pointed out by email to me two mistakes on the page Pr(infinity,1)Cat. I have fixed these now (I think).
The serious one was in the section Embedding into Cat where it said that $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits and colimits. But it only preserves limits. This is HTT, prop. 5.5.3.13. The wrong statement was induced from a stupid misreading of HTT, theorem. 5.5.3.18. Sorry.
The other mistake was that it said “full subcategory”. But of course by the very definition of $Pr(\infty,1)Cat$ if is not full in $(\infty,1)Cat$. I have fixed that, too, now.
The first mistake mentioned above is still claimed in the fourth paragraph at “As $\infty$-vector spaces”. I would fix it but I’m not sure what to make it say, since the subsequent paragraphs seem to be talking about colimits.
Thanks for catching this. I have removed the remaining “and colimits”. And for the time being I removed the statement about the action groupoid.
I added to Pr(infinity,1)Cat the definition of the tensor product, and some remarks about colimits in $Pr(\infty,1)Cat$ that coincide with limits, essentially because it is an $(\infty,2)$-category with local colimits.
I expanded the definition of the tensor product with an explanation of why accessibility is preserved on the right adjoints, which I think is something that Lurie glosses over a little too quickly.
By the way, the page Noncommutative Algebra claims that that paper is “subsumed as a part of monograph Higher Algebra”, but I can’t find this construction of the tensor product of locally presentable ∞-categories anywhere in Higher Algebra. Would it be better to say “mostly subsumed”?
What about section 4.8 of Higher Algebra?
Ah, there it is, thanks! The beginning of that section misled me about its contents.
Is it right that
$Ho\big(Pr(\infty,1)Cat\big) \;\simeq\; Ho\big(CombinatorialModelCat\big)$?
Something like this must be true. We have a bunch of entries that revolve around such a statement, but none that makes it actually explicit.
(Just asking a question, didn’t make any edits. Yet.)
Yes, surely, for a suitably definition of the RHS. The analogous statement for derivators was proven by Renaudin; I can’t recall offhand seing a statement of the $(\infty,1)$-categorical version.
Thanks! But shouldn’t plain 1-categorical localization of $CombinatorialModelCat$ at the Quillen equivalences work, too, and be equivalent to the plain 1-categorical homotopy category of $Pr(\infty,1)Cat$?
Yes. The “analogous statement for derivators” I referred to is an equivalence of homotopy 2-categories $Ho(LPrDer) \simeq Ho(CombModelCat)$.
Thanks, I have seen that. That’s why I was wondering if the 2-categorical level is necessary here to get an equivalence of the intended form.
Can one maybe easily get the analogous 1-categorical statement from the 2-categorical one? Is the 1-category obtained from a pseudo-localization by identifying isomorphic morphisms the corresponding 1-localization? This would be the statement of Renaudin’s Prop. 1.2.3, except that his $\pi_0$ seems to quotient out too much.
Oh yes, sorry, I didn’t understand what you were asking. You’re right that his $\pi_0$ is too strong in 1.2.3 (identifying any two arrows connected by a 2-morphism), but the theorem about the 1-categorical localizations is his 2.3.8. It’s an interesting question whether the homotopy 1-category (i.e. identifying isomorphic arrows) of the 2-categorical localization is always the same as the 1-categorical localization of the homotopy 1-category.
Ah, thanks, I had missed that corollary.
I’d have one more question now: Do we know which morphisms get inverted by this 1-categorical localization? Hopefully it is only the Quillen equivalences, not anything further.
I think that is somewhere in Renaudin’s paper too, though I don’t have time to check at the moment. However, doesn’t it also follow from the classical fact that a Quillen adjunction that induces an equivalence of homotopy categories is automatically a Quillen equivalence?
Right, thanks, it’s Prop. 2.3.4, and the proof is as you say. All right, great.
How about from the other end, is there no direct comparison between the 2-category of derivators and some homotopy 2-category of $\infty$-categories?
I see now where you added a pointer to Renaudin’s result in rev 29 of “derivator”, Your comment then indicates that you liked to think of this result as establishing the relation between derivators and $\infty$-categories in the first place. I just re-checked with Moritz’s thesis, and recall now that there is a similar attitude there. Hm.
How about a functor from $\infty$-categories to derivators? In our entry here it says that the evident construction “should” yield a derivator. Is this actually open?
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