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 $(Spec(A))^{top}$?). 



Geometric realization of Hochschild complex
Nice question. I did this computation a while ago, but I guess what you get, by taking the levelwise analytification of $X\_A$, is the canonical cosimplicial model for the (topological) free loop space of $(SpecA)^{top}$ (i.e. its totalization is homeomorphic to the free loop space $L((SpecA)^{top})$).


