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Sorry, if my notation is weird. Let K denote the monoidal homotopy category of chain complexes over k. Given K-functors $$F: C \otimes D ^{op} \to K, G: D \otimes E^{op} \to K,$$ what is the explicit description of their composition as profunctors? I believe the abstract description in terms of tracing out with coend, (which I don't understand yet) in this particular case has something to do with the bar complex, and in appropriate sense Hochschild homology.

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You can't actually do that composition with the homotopy category of chain complexes, since that doesn't have the limits and colimits required for the coend construction. If you do it in the $(\infty,1)$-category of chain complexes, with a homotopy coend, then you will get something to do with bar constructions and HH. – Mike Shulman Dec 28 '11 at 8:02
Thanks Mike, I think i actually needed K to be the dg category of chain complexes over k. Not sure that helps the completeness problem, but that seems closer related to $\infty,1$-category you mentioned. What would be a reference to learn this stuff. Specifically what exactly is the $\infty,1$-category you mentioned. Do you just define n-morphisms to be higher order chain homotopies, i.e. morphisms in the dg category of chain complexes? – yasha Dec 29 '11 at 18:59
The dg-category of chain complexes is cocomplete, so you can do composition there, but in general it won't be "homotopically meaningful" (i.e. represent the right construction at the $(\infty,1)$-category level) --- you'd need to do some sort of cofibrant replacement. I'd recommend learning some about model categories to start with; the $(\infty,1)$-category of chain complexes is probably best thought of as the homotopy $(\infty,1)$-category of the model category thereof. – Mike Shulman Jan 3 '12 at 2:38
Even if this answer comes after 3 years, I warmly recommend you to use Sasha Kuznetsov "Categorical Resolution of irrational singularities" as a quick reference for the language of DG-categories; using the enriched model for stable $(\infty,1)$-categories is the most natural thing to do if you want to do coends – Fosco Loregian Feb 24 '15 at 19:48
@MikeShulman: can you elaborate the connection between bar construction, homotopy coends and HH's in this setting? I stumbled upon this thread googling exactly these three terms – Fosco Loregian Feb 24 '15 at 19:49

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