Decompositions of the category of noncommutative sets and Hochschild and cyclic homology. (Q1407199)

The article builds up on work by \textit{T. Pirashvili} and \textit{B. Richter} [K-Theory 25, No. 1, 39-49 (2002; Zbl 1013.16004)] and gives a description of the cyclic homology \(HC_*(M)\) of a functor \(M\colon F(\text{ass})\to K\text{-Mod}\) and the Hochschild homology \(H_*(N)\) of a functor \(N\colon\Gamma(\text{ass})\to K\text{-Mod}\) as defined in [loc. cit.] as the cyclic homology \(HC_*(Mi_{(\Delta C)^{op}})\) of the cyclic module \(Mi_{(\Delta C)^{op}}\) respectively the homotopy \(\pi_*(N\widehat C)\) of the simplicial module \(N\widehat C\) in form of a special Tor-product. The categories \(F(\text{ass})\) and \(\Gamma(\text{ass})\) denote certain categories of noncommutative sets as also introduced in [loc. cit.]. The main result is the following: \[ HC_*(M)=\text{Tor}_*^{F(\text{ass})} (K_{\Sigma',F(\text{ass})},M)\quad H_*(N)=\text{Tor}_*^{\Gamma(\text{ass})} (K_{\Sigma',\Gamma(\text{ass})},N). \] Here \(K_{\mathcal{B,C}}\colon{\mathcal C}^{op}\to K\text{-Mod}\) is a functor defined for small categories \(\mathcal C\) which are representable as the composition of two categories \({\mathcal A}\circ{\mathcal B}\) where \(\text{Obj}({\mathcal A})=\text{Obj}({\mathcal B})=\text{Obj}({\mathcal C})\) and composition affects only the morphisms. The functor is explicitly defined by \(K_{\mathcal{B,C}}(c)=K(\coprod_{y \in{\mathcal C}}\text{Mor}_{\mathcal B}(c,y))\) where in general \(K(X)\) denotes the free \(K\)-module with base \(X\). The main result is obtained after showing that the two categories \(F(\text{ass})\) and \(\Gamma(\text{ass})\) can be represented in form of a decomposition as above.