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It seems to be well-known that the homotopy colimit of a simplicial chain complex (unbounded) can be computed by taking the totalization of the associated (half-plane) double complex. The totalization is what one gets out of the Bousfield-Kan formula, but chain complexes do not really form a simplicial model category and so it's not clear to me why this is automatically the right thing (at least levelwise cofibrant replacements are not needed since the result is homotopical).

However one can just consider the adjunction:

$$Tot:s\text{Ch} \rightleftarrows \text{Ch}: \underline{\text{Hom}}_{\text{Ch}}(C(\Delta^{\bullet}),-)$$ Both functors are homotopical and the right adjoint is naturally weakly equivalent to the diagonal, so this models the correct derived adjunction.

Now my question is, what about DGA's ? Can the homotopy colimit of a simplicial DGA be computed in the underlying category of chain complexes, i.e., via totalization ?

The comultiplication on $C(\Delta^{\bullet})$ equips the above right adjoint with a lax monoidal structure and hence it sends DGA's to simplicial DGA's. By modyfying the left adjoint one obtains a new adjunction $$Tot':s\text{DGA} \rightleftarrows \text{DGA}: \underline{\text{Hom}}_{\text{Ch}}(C(\Delta^{\bullet}),-)$$ where Tot' is defined by the coequalizer $$T(\text{Tot}(TX_\bullet)\rightrightarrows T(\text{Tot} X_\bullet)\rightarrow \text{Tot'} X_\bullet$$ Here $T$ denotes the tensor algebra (levelwise for simplicial objects) and the maps are induced by the T-algebra structure on $X_\bullet \in s\text{DGA}$ and the oplax monoidal structure on Tot. Now totalization should commute with the tensor algebra up to sufficiently nice chain homotopy equivalence so that one can conclude that Tot' is weakly equivalent to Tot. In particular it would model the homotopy colimit.

Is this really true and are there references for it ? (What about algebras over more general monads $T$ ?)

In the end i would just like to use this to identify the derived mapping space in DGA's (or algebras over a nice enough monad $T$) via the bar resolution:

$$map_{\text{DGA}}(X,Y)\simeq holim_{\Delta}map_{\text{Ch}}(T^\bullet X, Y)$$

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Conditions under which the answer is "yes", for a general monad $T$ (not just the free monoid monad) and a general model category (not just Ch(R)) are given in Batanin-Berger Tame Polynomial Monads, page 57. I don't really have time to figure out precisely what's needed for the case you're asking about, but they check their conditions for Ch(k) and Ch(R) on page 5. I suspect that everything is fine if you work over a field, and might be okay more generally over a commutative ring. This part of the paper builds on old work of Batanin: "Homotopy coherent category theory and $A_\infty$ structures", which might contain the special case you're looking for. Or you can just check the conditions to deduce it from the general case in the tame paper.

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  • $\begingroup$ I'll have a look at that paper, thanks. $\endgroup$
    – Christian Wimmer
    Commented Feb 25, 2017 at 8:59
  • $\begingroup$ Are you saying the statement about homotopy colimits can be extracted from Thm 8.2 or are you referring to what they do with bar resolutions in the proof ? $\endgroup$
    – Christian Wimmer
    Commented Feb 26, 2017 at 13:44
  • $\begingroup$ The proof is what I was referring to. It's a vast generalization of Batanin's work in that $A_\infty$ paper, and seems to be the state of the art for questions like this. $\endgroup$
    – David White
    Commented Feb 26, 2017 at 15:18

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