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Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the homology of the geometric realization of this quasi-category (in particular a simplicial set) should be related to Hochschild homology of A. Is anything known about this?

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    $\begingroup$ Have you at least thought whether the coefficients for both homologies could be the same kind of object? $\endgroup$
    – Fernando Muro
    Commented May 14, 2013 at 14:00
  • $\begingroup$ The negative Hochschild cohomology groups are realized as homotopy groups of the classifying space ($\infty$-groupoid) of dg categories. See Corollary 8.5 of Toen's The homotopy theory of dg categories and derived Morita theory. $\endgroup$
    – Chris Brav
    Commented May 14, 2013 at 20:05
  • $\begingroup$ @Chris, I think Yasha asks about $HH_\ast(C)$, not $HH^\ast$. Yasha, I see more naturally a relationship between $HH_\ast$ and the homotopy groups of the (free loop space of the) geometric realization; if you want to relate it to homology, you probably want some Hurewicz type assumption on the nerve of the category. Dually, the result Chris cites is fairly intuitive--$HH^\ast$ is endomorphisms of the identity of $C$, so you can see it at the $\pi_2$ level of the (largest) oo-groupoid contained in the oo-category of dg-categories, with basepoint $C$. $\endgroup$
    – Hiro Lee Tanaka
    Commented May 14, 2013 at 22:29
  • $\begingroup$ @Fernando, I mean of course $\mathbb{Z}$ coefficients for homology. Both are naturally Z modules if you wish :) @Hiro, thanks I also realized later by looking at a very specific example that one should instead consider the free loop space of the geometric realization. Although I am not yet convinced on homotopy groups vs homology groups, in the example I was considering it didn't matter as the realization of the dg-nerve had the homotopy type of the circle. If there is demand I can add this example to the body of question. $\endgroup$
    – yasha
    Commented May 15, 2013 at 17:44
  • $\begingroup$ @yasha what's Hochschild homology of a dg-category with coefficients in $\mathbb Z$? $\endgroup$
    – Fernando Muro
    Commented May 16, 2013 at 7:14

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