  The pull back of a flat bundle $E\rightarrow X$ along the evaluation map
$\pi: \mathcal{L} X \to X$ from the free loop space $\mathcal{L} X$ to $X$
comes equipped with a canonical automorphism given by the holonomies of $E$.
This construction naturally generalizes to flat $\mathbb{Z}$-graded connections
on $X$. Our main result is that the restriction of this holonomy automorphism
to the based loop space $\Omega_* X$ of $X$ provides an $A_\infty$
quasi-equivalence between the dg category of flat $\mathbb{Z}$-graded
connections on $X$ and the dg category of representations of
$C_\bullet(\Omega_* X)$, the dg algebra of singular chains on $\Omega_* X$.
