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I'm looking for a reference for the following fact:

take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough projectives, although I'm willing to add more hypotheses, because the $\mathcal A$ I want to use it for is the category of connective modules over some connective dga ); then a model for the homotopy colimit of $X$ is the total complex of the bicomplex associated to $X$.

I know a proof (I haven't checked the details so I'm not sure it works for an arbitrary dga - it works at least for discrete rings), so that's not what I'm looking for (except if you have an especially short and elegant one, then it wouldn't hurt to see it); I'm mostly looking for a reference.

I know the result is mentioned in Dugger's A primer on homotopy colimits (proposition 19.9) but there seems to be no proof in there - so I'll add the criterion that the reference should contain a proof.

This may be related to this question, which relates the total complex and the diagonal - since there is an answer with a reference there, it would also suffice to provide a reference for the fact that the diagonal is a model for the homotopy colimit (actually, this would be enough for other reasons : one can use the diagonal model for simplicial objects that land in $\mathcal A$, and then use homotopy cofinality of $\Delta^{op}\to \Delta^{op}\times \Delta^{op}$ to get the result for an arbitrary simplicial chain complex).

For the latter, I know references for simplicial sets, but not for simplicial $\mathcal A$-objects (and in the case of a discrete ring, one may use this as well via the usual adjunction).

The answers given here seem to be unsatisfactory given the comments below.

Here, the question itself provides a sketch of proof for $\mathbb Z$ which I think can be adapted to the general case, but the adjunction that is mentioned does not seem crystal clear to me (if you could explain it, that would also be great) and it's not a reference.

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  • $\begingroup$ This is a duplicate of mathoverflow.net/questions/194010/…, which is already referenced in the main post. $\endgroup$
    – Dmitri Pavlov
    May 22, 2020 at 23:12
  • $\begingroup$ @DmitriPavlov: you're right but the answers there didn't seem to solve my problem so I wasn't sure what the etiquette was in that situation $\endgroup$
    – Maxime Ramzi
    May 23, 2020 at 8:36

2 Answers 2

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See Problem 4.23 and Problem 4.24 (with proofs) of Ulrich Bunke's Differential cohomology.

The underlying abstract machinery for computing homotopy (co)limits via homotopy (co)ends is presented by Sergey Arkhipov and Sebastian Ørsted in Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories.

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  • $\begingroup$ Thank you for your answer ! However it seems that the first reference only deals with $\mathcal A = \mathbf{Ab}$ (as far as I can tell + the proof 4.24. uses a claim that I would typically prove using the desired result, that is "every element in $Ch^{\Delta^{op}}$ is a homotopy colimit of a diagram of $c(\mathbb Z)$'s " ). For the second one, I think what I'm missing is an understanding of $Tot$ as a coend (or as an end, for the dual version) - do you maybe have references for that ? $\endgroup$
    – Maxime Ramzi
    May 23, 2020 at 8:40
  • $\begingroup$ @MaximeRamzi: There is nothing special about Z, the proof works in the same way for any locally presentable abelian category. Also, 4.23 works in the opposite category, giving you the desired presentation of a colimit as a coend. $\endgroup$
    – Dmitri Pavlov
    May 23, 2020 at 15:29
  • $\begingroup$ As for your question about presenting totalization as a 1-coend, this is a purely 1-categorical question. It suffices to observe that chains on Δ^n only have a single summand (Z in degree n) that does not come from Δ^k for k<n. This is precisely the copy of Z responsible for totalization via shifting in degree by n. I have never seen this (simple) argument proved explicitly in the literature. $\endgroup$
    – Dmitri Pavlov
    May 23, 2020 at 15:32
  • $\begingroup$ well, the proof can certainly go through in an arbitrary locally presentable abelian category, but as it is written, it does seem to use some stuff from $\mathbb Z$ (specifically the chain complex $C_*(\Delta^n)$ and its equivalence to $\mathbb Z$, that is, you need some monoidal unit or something) ; and since I'm looking for a reference, I'd like one that explicitly states that it works for an arbitrary good enough abelian category (although if I don't find one, I'll probably end up citing that one and saying "it goes through similarly in more generality"). For the coend, yes you're right ! $\endgroup$
    – Maxime Ramzi
    May 23, 2020 at 15:35
  • $\begingroup$ (of course I'm not claiming that you can't adapt $C_*(\Delta^n)$ to an arbitrary abelian category, I'm just saying that as written, it is not that general) $\endgroup$
    – Maxime Ramzi
    May 23, 2020 at 15:37
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I wrote a note for referential purposes. I hope this helps.

Arakawa, K. (2023). Homotopy Limits and Homotopy Colimits of Chain Complexes. arxiv.2310.00201

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