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It seems to be a well-known fact that homotopy (co)limits of (co)simplicial diagrams of nonnegatively graded (co)chain complexes in (Grothendieck) abelian categories can be computed by using the Dold-Kan correspondence to pass to double complexes and then applying the totalization functor for double complexes, possibly applying the truncation functor afterward. For example, this is claimed (without proof) in Section 16.8 of Dugger's A primer on homotopy colimits.

In the above, “homotopy (co)limit” is used in the abstract ∞-categorical sense, i.e., the homotopy terminal (respectively initial) object in the ∞-category of (co)cones. It can be presented as the appropriately derived functor of the ordinary (co)limit functor in the setting of (stable) model categories or as the quasicategorical (co)limit in the setting of stable quasicategories.

Is there a written proof of this result in the literature?

What about the case of unbounded chain complexes?

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  • $\begingroup$ Totalisation (by $\bigoplus$) preserves filtered colimits, and filtered colimits are homotopical, so you can reduce the unbounded case to the bounded case. No? $\endgroup$
    – Zhen Lin
    Commented Jan 15, 2015 at 16:54
  • $\begingroup$ @ZhenLin: I guess so, at least for homotopy colimits. What about homotopy limits of unbounded complexes? $\endgroup$
    – Dmitri Pavlov
    Commented Jan 15, 2015 at 18:10
  • $\begingroup$ The Dold-Kan correspondence I'm familiar with needs a simplicial object, are you only talking about homotopy colimits indexed by $\Delta^{\mathrm{op}}$? $\endgroup$
    – Omar Antolín-Camarena
    Commented Jan 15, 2015 at 20:07
  • $\begingroup$ @OmarAntolín-Camarena: Yes, I forgot to mention that the diagrams are (co)simplicial. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 15, 2015 at 21:26
  • $\begingroup$ Meanwhile, a more complete answer is available here: mathoverflow.net/questions/361064/… $\endgroup$
    – Dmitri Pavlov
    Commented Jul 26, 2020 at 22:41

3 Answers 3

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I wrote a note for referential purposes. I hope that this will be helpful.

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

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Rodríguez González, Beatriz(E-CSIC-IM) Simplicial descent categories. (English summary) J. Pure Appl. Algebra 216 (2012), no. 4, 775–788

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    $\begingroup$ I guess these results must be combined with Theorem 3.1 in “Realizable homotopy colimits” by the same author, which proves that simplicial descent categories give rise to homotopy colimits in the usual sense. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 16, 2015 at 21:37
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The case of unbounded (cochain) complexes and colimits is treated with great detail in the paper:

Alonso Tarrío, Leovigildo; Jeremías López, Ana; Souto Salorio, María José Localization in categories of complexes and unbounded resolutions. Canad. J. Math. 52 (2000), no. 2, 225–247.

(Sorry for the self-reference). The theory is developed in detail in section 2 of this paper but it is not checked explicitly that they correspond to the usual homotopy colimits by the Dold-Kan correspondence.

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  • $\begingroup$ How do you know that the construction that you describe in your paper has the universal property required of homotopy colimits? $\endgroup$
    – Dmitri Pavlov
    Commented Jan 15, 2015 at 16:49
  • $\begingroup$ Strictly speaking, I don't know. However the fact that it is functorial and gives a complex quasi-isomorphic to the true colimit gives a hint in this direction. In any case, it should not be difficult to prove it. $\endgroup$
    – Leo Alonso
    Commented Jan 15, 2015 at 17:04
  • $\begingroup$ I certainly agree that it's not difficult to prove, but I would like to avoid writing up something that seems to be so classical. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 15, 2015 at 18:09

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