Cubical vs. simplicial Hochschild cohomology 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?
 A: 
What are the relative advantages and disadvantages between the simplicial >and cubical theories?

Dear Emily,
For Hochschild homology I cannot say nothing because I have never worked on it. But in the level of higher category theory I published some results which show that we have algebraic (and operadic) approach of weak cubical $\infty$-categories similar to globular higher category theory. My results on cubical higher operads are not available now (this was in the site of ihes, but they disappear for reasons I ignore). My feeling is the fact that cubical world is something between globular and simplicial. Having an algebraic version of cubical weak $\infty$-groupoids is a clear advantage compare to simplicial, where algebraic approach exists but are less "algebraic" than for the cubical geometry, in the sense of there is a monad on cubical sets which algebras are cubical weak higher groupoids, but this seems not the case for simplicial. Here are some references about my work on cubical things (If ihes put again my approach of cubical operads, I will let you know):
https://arxiv.org/pdf/2102.09787.pdf
https://cgasa.sbu.ac.ir/article_101544_331b33b547e4eadc6d51fc63f0c9c84c.pdf
https://cgasa.sbu.ac.ir/article_101533_04e2a777ec95ba4c14b12d599ae71c2e.pdf
Also Ronald Brown told me many times that computations are easier with cubical shapes, especially because for these shapes he got Van Kampen theorem for cubical strict higher groupoids.
A: Dear Emily (I prefer to write here, because it gives me more space...). You are welcome :-) Actually my work between 2018--2021 has shown that we have now the following results on cubical higher category: 1) There is a monad on CSets, the category of cubical sets, which algebras are cubical weak $\infty$-categories (with or without connections), a monad on CSetsxCSets which algebras are cubical weak $\infty$-functors (with or without connections) a monad on CSets^4 which algebras are cubical weak $\infty$-natural transformations (with or without connections) a monad on CSets which algebras are cubical weak $\infty$-groupoids (with or without connections)2) The monad of cubical strict $\infty$-categories (with or without connections) is cartesian (the proof is quite hard but very pretty!), thus leads to a cubical operads, analogue to the globular operads of Batanin, which algebras are cubical weak $\infty$-categories (with or without connections). Also in my ihes preprint (which disappears ! But I am going to contact soon IHES in order they put it again in their site), I built a fundamental weak $\infty$-groupoid functor from the category Top to the category of weak $\infty$-groupoids; a last result related to my combinatorial work on cubes is a nice definition of cubical weak $\infty$-groupoids as model of a cubical coherators, where globular coherators are kind of projective sketches invented by Grothendieck. Also I described a cubical coherator (here: https://arxiv.org/pdf/2102.09787.pdf) which sets-models are cubical $\infty$-categories (with or without connections). Again what surprise me, is the important fact that in one hand there is a lot of analogies between simplicial and cubical constructions (as you discussed above), but also a complete analogy with globular higher categories where algebraic constructions are possible, and cubical higher categories. This is the reason why I wrote that, for me, cubical appears as a bridge between globular and simplicial. I hope in the Future we will able to link cubical homology and algebraic cubical higher categories.
