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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 26th 2012

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.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeFeb 14th 2017

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 14th 2017

Thanks for catching this. I have removed the remaining “and colimits”. And for the time being I removed the statement about the action groupoid.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeFeb 16th 2017

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.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeFeb 19th 2017

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”?

• CommentRowNumber6.
• CommentAuthorCharles Rezk
• CommentTimeFeb 19th 2017

What about section 4.8 of Higher Algebra?

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeFeb 20th 2017

Ah, there it is, thanks! The beginning of that section misled me about its contents.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 5th 2018
• (edited Jul 5th 2018)

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.)

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJul 5th 2018

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJul 5th 2018

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 $Pr(\infty,1)Cat$?

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeJul 5th 2018

Yes. The “analogous statement for derivators” I referred to is an equivalence of homotopy 2-categories $Ho(LPrDer) \simeq Ho(CombModelCat)$.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 6th 2018

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.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeJul 6th 2018

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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJul 6th 2018

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.

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeJul 6th 2018

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?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJul 7th 2018

Right, thanks, it’s Prop. 2.3.4, and the proof is as you say. All right, great.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJul 7th 2018

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?