Twisted Simplicial Groups and Twisted Homology of Categories

Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ 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 $δ\colon V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto δ_v$ such that $δ_v\circ δ_w=δ_w\circ δ_v$ if there exists an edge in $A$ joining $v$ with $w$ or an arrow either from $v$ to $w$ or from $w$ to $v$. 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.

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