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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2013

    created dg-nerve

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

    added publication data and warning about differing numbering to this item

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    Question: Given a dg-category C\mathbf{C} with underlying ordinary category CC: (when) is the dg-nerve of C\mathbf{C} equivalent (as a quasi-category) to the homotopy-coherent nerve of the simplicial localization of CC at the canonical weak equivalences?

    I see that Rem. 1.3.1.1 in Higher Algebra (p. 83) says this for homotopy categories (only).

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

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

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

    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 ?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    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 sSetsSet-functor model category

    sFunc(𝒢(X),sCh •(k)) 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.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2023

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

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    Yes, that’s exactly it. Thanks!!

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

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    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…

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

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

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

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

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

    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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023
    • (edited May 7th 2023)

    I see, thanks.

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

    diff, v11, current

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeMay 7th 2023

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

    • CommentRowNumber15.
    • CommentAuthorTim_Porter
    • CommentTime1 day ago

    Added the earliest reference to this that I know of.

    diff, v12, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTime1 day ago

    have adjusted a little and copied the item also to the author’s pages

    diff, v14, current

    • CommentRowNumber17.
    • CommentAuthorTim_Porter
    • CommentTime1 day ago

    Thanks. I was a bit rushed yesterday, so did not get around to adjusting those pages.

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