  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 $$ \delta\colon
V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto \delta_v $$ 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.
