Let $M$ be a "good" topological space such as a manifold or a CW complex and assume it's connected. We use $\operatorname{Loc}(M)$ to denote the category of local systems either as a dg category over a field $k$ or a stable category. For this, we mean either representations from the fundamental $\infty$-groupoid $\pi_\infty(M)$ to the coefficient category or the category consists of locally constant sheaves of the same coefficient.
Now pick any point $x \in M$ and one can consider the based loop space $\Omega_x M$ over $x$. This is a group object in spaces by concatenation and thus its chains $C_* (\Omega_x M )$ is a ring. I have the impression that there is the equality $$\operatorname{Loc}(M) = C_* (\Omega_x M) - \operatorname{Mod}.$$ My questions is where I can find a reference or is there a simple argument of it? I can image, by the point of view of Morita theory, that one can write down a local system which is a compact generator and whose endomorphism is $C_* (\Omega_x M)$, maybe simply the local system which assigns a point $y$ to the ring $C_* (\Omega_y M)$? But I'm confused with the details.
This is an example of Koszul duality. It's probably somewhere in Lurie, but I couldn't pinpoint it, so here's another reference - lemma 3.9 is what you want.