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Question: Given a dg-category $\mathbf{C}$ with underlying ordinary category $C$: (when) is the dg-nerve of $\mathbf{C}$ equivalent (as a quasi-category) to the homotopy-coherent nerve of the simplicial localization of $C$ at the canonical weak equivalences?
I see that Rem. 1.3.1.1 in Higher Algebra (p. 83) says this for homotopy categories (only).
Just a moment, I will provide the new reference in a moment which seems to/may help with this question.
We describe several equivalent models for the infinity-category of infinity-local systems of chain complexes over a space using the framework of quasi-categories. We prove that the given models are equivalent as infinity-categories by exploiting the relationship between the differential graded nerve functor and the cobar construction. We use one of these models to calculate the quasi-categorical colimit of an infinity-local system in terms of a twisted tensor product.
P.S. you are in characteristic zero, I suppose ?
I am aware of this article, but i don’t see that it is close to addressing what I am asking.
I am asking because there is the other model for $\infty$-local systems which I have been discussion (with Dmitri) around here, namely presented by the $sSet$-functor model category
$sFunc\big(\mathcal{G}(X),\, sCh_\bullet(k)\big)$into a simplicial enhancement of the model structure on chain complexes.
To show that this is equivalent to Block & Smith-model via dg-nerves and its variants discussed by Rivera & Zeinalian one needs something like a positive answer to the above question.
Re #3: Does Proposition 5.17 in the paper https://arxiv.org/abs/1602.01515 answer your question?
Yes, that’s exactly it. Thanks!!
And it does apply to $Ch_\bullet(k)$, great.
Ah, but as your paper points out, the case of chain complexes that I am after is also Prop. 1.3.4.5 in Higher Algebra, had missed that. Okay, all the better, I’ll record both statements on our page now…
(Dmitri, do you really insist to restrict to small categories in that Prop. 5.17?)
Re #10: The smallness condition is removed when 5.17 is deployed in Theorem 5.33. See also Remark 5.32.
To generalize 5.17 to non-small dg-categories C, you would have to replace the model category C-Mod with the Chorny–Dwyer model category of small dg-presheaves on C.
I guess at the time I wasn’t familiar with the Chorny–Dwyer model structure yet, which explains why I stated 5.17 for small dg-categories.
I see, thanks.
On another note, I have added publication data for this item:
I see. Thanks for the discussion, it is useful to me as well.
Thanks. I was a bit rushed yesterday, so did not get around to adjusting those pages.
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