Twisted Simplicial Groups and Twisted Homology of Categories

Let $A$ be either a simplicial complex $K$ or a small category $C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $\delta$ from vertices to endomorphisms of $G$ with certain commutativity condition. We give a canonical construction of twisted simplicial group as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy type of of this simplicial group as the loop space over certain twisted smash product.