I see. Thanks for the discussion, it is useful to me as well.

]]>I see, thanks.

On another note, I have added publication data for this item:

- Giovanni Faonte,
*Simplicial nerve of an $\mathcal{A}_\infty$-category*, Theory and Applications of Categories**32**2 (2017) 31-52 [arXiv:1312.2127, tac:32-02]

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.

]]>Okay, I have recorded these statements in a new section “Properties – Compatibility with the simplicial nerve” (now here)

]]>(Dmitri, do you really insist to restrict to *small* categories in that Prop. 5.17?)

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…

Yes, that’s exactly it. Thanks!!

And it does apply to $Ch_\bullet(k)$, great.

]]>Re #3: Does Proposition 5.17 in the paper https://arxiv.org/abs/1602.01515 answer your question?

]]>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.

]]>- Manuel Rivera, Mahmoud Zeinalian,
*The colimit of an infinity local system as a twisted tensor product*, Higher Structures 4(1), (2020) mpg:pdf arXiv:1805.01264

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 ?

]]>Just a moment, I will provide the new reference in a moment which seems to/may help with this question.

]]>**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).

added publication data and warning about differing numbering to this item

- Jonathan Block, Aaron M. Smith, Def. 2.3 (2.4 in the preprint) of:
*The higher Riemann–Hilbert correspondence*, Advances in Mathematics**252**(2014) 382-405 [arXiv:0908.2843, doi:10.1016/j.aim.2013.11.001]

created *dg-nerve*