TITLE: Geometric realization of Hochschild complex
QUESTION [10 upvotes]: Let $A$ be a commutative $\mathbb{C}$-algebra, and consider $C_{\bullet}(A,A)$ the simplicial Hochschild homology module of $A$ with respect to itself (i.e. $C_{n}(A,A)=A^{\otimes (n+1)}$). This is a simplicial commutative $\mathbb{C}$-algebra and we can take its $Spec$ levelwise, to get a cosimplicial $\mathbb{C}$-scheme $X_A :=Spec(C_{\bullet}(A,A))$. If $A$ is of finite type over $\mathbb{C}$, then we may take levelwise the associated topological space for the analytic topology, and get a cosimplicial topological space, denoted $X_{A}^{top}$. My question is whether somebody knows what the geometric realization of $X_{A}^{top}$ looks like (e.g. any relation to the topological free loop space of $(Spec(A))^{top}$?).

REPLY [3 votes]: Sam Gunningham had the right idea, but I would like to fix it up for the construction of the homotopy pullback that I am aware of.
Alternatively one can compute $HH(A;A)$ as $\mathrm{Tor}^{A^e}(A,A)$ where $A^e$ is the enveloping algebra of $A$, $A\otimes A^{op}$, since your $A$ is commutative $A^{e}\cong A\otimes A$. These Tor groups can be calculated via the simplicial complex which in degree $n$ is $A\otimes (A^{e})^{\otimes n}\otimes A\cong A^{\otimes {2n+2}}$. As a side note the geometric realization of this complex gives a topological commutative ring (Because $A$ is commutative) which models $THH^{HZ}(HA;HA)$ via the Eilenberg-Maclane functor. Applying your contravariant functor to spaces to the simplicial algebra we obtain a cosimplicial space modeling the homotopy pullback $Spec(A)^{top}\times_{Spec(A)^{top}\times Spec(A)^{top}}\times Spec(A)^{top}$ (see 3.3 of http://www.math.uiuc.edu/~reldred2/tot-primer.pdf ) which as Sam pointed out, models the free loop space.