The pull back of a flat bundle |$E\rightarrow X$| along the evaluation map |$\pi: \mathcal{L} X \rightarrow X$| 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 |$\mathsf{A}_\infty$| 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 |$\Omega_* X$|⁠.