Twisted Simplicial Groups and Twisted Homology of Categories

Abstract

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 $$\delta :V(A)⟶\mathrm{End}(G),v\mapsto {\delta }_v$$ such that ${\delta }_v\circ {\delta }_w={\delta }_w\circ {\delta }_v$ 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.

19 pages