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:
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 , 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 -local systems which I have been discussion (with Dmitri) around here, namely presented by the -functor model category
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.
]]>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 with underlying ordinary category : (when) is the dg-nerve of equivalent (as a quasi-category) to the homotopy-coherent nerve of the simplicial localization of 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
created dg-nerve
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