Let $G$ be a group, and let $L G = B G^{S^1}$ be the functor-groupoid into $B G$ out of $S^1 = B\mathbb{Z}$. So the objects of $L G$ are elements of $G$ and its morphisms are conjugations. An $n$-simplex in the nerve of $L G$ is a string of $n$ conjugations, which means it is determined by $n+1$ elements of $G$. Therefore, the free abelian group on the $n$-simplices in the nerve of $L G$ is isomorphic to $\mathbb{Z}[G^{n+1}] \cong \mathbb{Z}[G]^{\otimes (n+1)}$. This strongly suggests that the chain complex obtained from the nerve of $L G$ should be the Hochschild complex of $\mathbb{Z}[G]$, but making this identification extend to all the face maps seems to require a clever choice of the isomorphism between the $n$-simplices in $N(L G)$ and $G^{n+1}$, and I havenâ€™t yet managed to be sufficiently clever. Does this exist anywhere? Does anyone know the answer?

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