Simplicial Hochschild cohomology.
$\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\B}{\mathrm{B}}\newcommand{\Obj}{\mathrm{Obj}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Mod}{\mathsf{Mod}}$One way to define the Hochschild cohomology of an $R$-algebra $S$ with coefficients in a bimodule $M$ is to first consider $S$ as a $\Mod_R$-enriched category $\B S$ and then define a simplicial $R$-module $\HH^\bullet_{R,\triangle}(S;M)$, which may be pictured as follows: \begin{align*} \prod_{A\in\Obj(\B S)}M &\rightrightarrows \prod_{A,B\in\Obj(\B S)}\Mod_R(\Hom_{\B S}(A,B),M)\\ &\underset{\to}{\rightrightarrows}\prod_{A,B,C\in\Obj(\B S)}\Mod_R(\Hom_{\B S}(A,B)\otimes_R\Hom_{\B S}(B,C),M)\\ &\underset{\rightrightarrows}{\rightrightarrows}\cdots \end{align*} (Note that since $\B S$ has only a single object, this reduces to the more simple form \begin{align*} M &\rightrightarrows \Mod_R(S,M)\\ &\underset{\to}{\rightrightarrows}\Mod_R(S\otimes_RS,M)\\ &\underset{\rightrightarrows}{\rightrightarrows}\cdots, \end{align*} but the first form displayed above is the one that leads to a notion of Hochschild homology for arbitrary $R$-linear categories, and highlights the relation to the simplicial nerve (see below).)
Looking at $\HH^\bullet_{R,\triangle}(S;M)$, we see that it is in a sense a way to "integrate" $M$ against the simplicial nerve of $\B S$, whose $R$-module of $n$-simplices is given by $$\mathrm{N}_{n}(\B S)=\coprod_{A_1,\ldots,A_n\in\Obj(\B S)}\Hom_{\B S}(A_1,A_2)\otimes_R\cdots\otimes_R\Hom_{\B S}(A_{n-1},A_n).$$
Cubical Hochschild cohomology
Now, it seems to me that one could just as well build a cubical $R$-module $\HH^\bullet_{R,\square}(S;M)$, using the cubical nerve of $\B S$ instead. Its first two terms would be the same as $\HH^\bullet_{R,\triangle}(S;M)$, but the third one would be the quotient of $$\prod_{A,X,Y,B\in\Obj(\B S)}\Mod_R(\Hom_{\B S}(A,X)\otimes_R\Hom_{\B S}(X,B)\otimes_R\Hom_{\B S}(A,Y)\otimes_R\Hom_{\B S}(Y,B),M)$$ where we identify the compositions \begin{align*} \Hom_{\B S}(A,X)\otimes_R\Hom_{\B S}(X,B) &\to \Hom_{\B S}(A,B),\\ \Hom_{\B S}(A,Y)\otimes_R\Hom_{\B S}(Y,B) &\to \Hom_{\B S}(A,B), \end{align*} just like how we may write the third term of $\HH^\bullet_{R,\triangle}(S;M)$ as the quotient of $$\prod_{A,B,C\in\Obj(\B S)}\Mod_R(\Hom_{\B S}(A,B)\otimes_R\Hom_{\B S}(B,C)\otimes_R\Hom_{\B S}(A,C),M)$$ where now we identify the image of the composition $$\Hom_{\B S}(A,B) \otimes_R\Hom_{\B S}(B,C) \to \Hom_{\B S}(A,C)$$ with the $\Hom_{\B S}(A,C)$ in $$\prod_{A,B,C\in\Obj(\B S)}\Mod_R(\Hom_{\B S}(A,B)\otimes_R\Hom_{\B S}(B,C)\otimes_R\Hom_{\B S}(A,C),M).$$
Both of the quotients above come from the form of:
- The $2$-simplices of the simplicial nerve of a category, i.e. a triangle $g\circ f\sim h$
vs.
- The $2$-cubes of the cubical nerve of a category, i.e. a square $g\circ f\sim k\circ h$.
Questions.
Has anyone tried developing such a theory of "cubical Hochschild homology", carrying a similar relation to usual Hochschild homology as cubical vs. simplicial singular homology?
Do the cubical homotopy groups of the cubical $R$-module $\HH^\bullet_{R,\square}(S;M)$ agree with the simplicial homotopy groups of the simplicial $R$-module $\HH^\bullet_{R,\triangle}(S;M)$?
What are the relative advantages and disadvantages between the simplicial and cubical theories?
(Apart from the fact that the annoying quotienting process in $\HH^\bullet_{R,\square}(S;M)$ can be skipped in $\HH^\bullet_{R,\triangle}(S;M)$.)
The Hochschild $n$-cocycles of the complex of $R$-modules associated to $\HH^\bullet_{R,\triangle}(S;M)$ are related to the deformation theory of $S$ and $M$, at least in low degrees. Do we have a similar relation for cubical Hochschild cohomology as defined above, using cubical Dold–Kan?
