Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where $$ C_n(A):=A^{\otimes n+1} $$ and the differntial $b: C_n(A)\rightarrow C_{n-1}(A)$ is defined to be $$ b(a_0\otimes\ldots\otimes a_n):=\sum_{i=0}^{n-1}(-1)^i a_0\otimes \ldots\otimes a_ia_{i+1}\otimes\ldots\otimes a_n. $$ The Hochschild homology $HH^{\cdot}(A)$ is defined to be the homology of the complex $(C_{\cdot}(A),b)$.

Moreover, $C_{\cdot}(A)$ is a simplicial module where $d_i: C_n(A)\rightarrow C_{n-1}(A)$ is defined to be $$ d_i(a_0\otimes\ldots\otimes a_n):=a_0\otimes \ldots\otimes a_ia_{i+1}\otimes\ldots\otimes a_n $$ and $b=\sum_{i=0}^{n-1}(-1)^id_i$ according to standard simplicial homology theory.

On the other hand, we have a simplicial model of $S^1$, denoted by $S^1_{\cdot}$ (see Loday's Free loop space and homology). Where $$ S^1_0=\{*\}\\ S^1_1=\{s_0(*),\tau\} $$ where $d_0\tau=*$ and $d_1\tau=*$. For $n>1$ $$ S^1_n=\{s_0^n(*),s_{n-1}\ldots\widehat{s_{i-1}}\ldots s_0\tau,i=1,\ldots,n\}. $$ We see that each $S^1_n$ consists of $n+1$ elements. Anyway, it is not difficult to check that its geometric realization $|S^1_{\cdot}|$ is homeomorphic to $S^1$.

$\textbf{If the algebra A is commutative}$, the Hochschild chain complex $C_{\cdot}(A)$ can be considered as the tensor product of $S^1_{\cdot}$ and $A$, and the $d_i$'s in $C_{\cdot}(A)$ are induced by the $d_i$'s in $S^1_{\cdot}$.

For example, both $d_0$ and $d_1$ have to map the two elements in $S^1_1$ to the same element $*$ in $S^1_0$ and both of them correspond to $$ a_0\otimes a_1\mapsto a_0a_1. $$ Here we don't need to distinguish $a_0a_1$ and $a_1a_0$ since $A$ is commutative.

$\textbf{My question}$ is: when $A$ is NOT commutative, could we also get the simplicial module $C_{\cdot}(A)$ from $S^1_{\cdot}$?


The article "Hochschild and cyclic homology via functor homology," by Pirashvili and Richter, gives precisely such a description.

In this paper they describe a category of "noncommutative sets," isomorphic to one described earlier by Fiedorowicz and Loday. The objects are finite sets, and maps are maps of finite sets together with a total order on each preimage. Associative algebras are not tensored over finite sets, but they are tensored over noncommutative sets, and if the algebra is commutative this agrees with the tensor over the underlying set. They then show that the simplicial set $S^1_\bullet$ lifts to the structure of a simplicial noncommutative set, and the Hochschild complex of an associative algebra $A$ comes from the levelwise tensor of $A$ with $S^1_\bullet$.


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