The n-Category Café

Re: Hochschild Homology As Cohomology of Loop Space Objects

The following is something I kept polishing the discussion of.

First a word on where we come from:

in an $\infty$-topos $\mathbf{H}$, cohomology of an object $X$ with coeffients in some object $A$ is just the external hom $\mathbf{H}(X,A)$.

Specifically, Hochschild cohomology of an object $X$ is the cohomology of its $\infty$-powering $X^{S^1}$ with the categorical circle

$$S^1 \simeq * \coprod_{* \coprod *} * \,.$$

Dually, if we have a nice assignment $\mathcal{O}$ of function $\infty$-algebras $\mathcal{O}(X)$ to objects $X$, and if $X$ is $\mathcal{O}$-perfect in that taking functions commutes with taking $\infty$-limits, then we have that the Hochschild homology complex of $X$ is

$$\mathcal{O}(X^{S^1}) \simeq S^1 \cdot \mathcal{O}(X) \,,$$

where now on the left we have the $\infty$-copowering of $\infty$-algebras over $\infty$-groupoids.

More generally, for $K$ any $\infty$-groupoid and $X$ $\mathcal{O}$-perfect, we say that

$$\mathcal{O}(X^K) \simeq K \cdot \mathcal{O}(X)$$

is the higher order Hochschild homology of $X$ with respect to $K$.

So that’s the abstract story. Now we want to unwind this and break it down to 1-categorical algorithms to actually compute things. If we choose our 1-categorical models of the $\infty$-categories well enough, then there is a chance that this $\infty$-categorical copowering over $\infty$-groupoids is modeled by an ordinary copowering over the category of simplicial sets of an ordinary catergory of algebras.

This is in fact an old observation by Pirashvili: he and his school never seem to talk eplicitly about the notion of copowering of algebras over simplicial sets, but that’s precisely what people write down when they talk about higher order Hochschild homology.

Specifically, just recently Ginot, Treidler, Zeinalian put the notes that had been sitting on Grégory’s website onto the arXiv. In these notes, they consider higher order Hochschild homology in terms of (implicitly) copowering of dg-algebras over simplicial sets and show that this is homotopy-good in the first argument, and that this effectively characterizes it.

They don’t say explicitly that this implies that the derived $\infty$-functor of that 1-categorical copowering is indeed the $\infty$-categorical copowering, but it is kind of obvious that this is implied. I tried to think up an alternative proof here, but it needs a bit more attention. However I have to call it quits now and go home.