I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of derived flat bundles where the fibers are complexes of vector spaces instead, and monodromy gives quasi-isomorphisms of complexes? Is there one for principal $G$-bundles?
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2$\begingroup$ There is a notion of $\infty$-local system which sounds like what you are looking for. There are several questions about this notion: mathoverflow.net/questions/153344 mathoverflow.net/questions/403734 mathoverflow.net/questions/438295 mathoverflow.net/questions/356749 $\endgroup$– Dan PetersenCommented Feb 5 at 10:32
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Maybe not exactly what you are looking for, but perhaps of relevance are Flat super-connections. That let $E\to B$ be a $\mathbb{Z}_2$ graded vector bundle, and $A=d_0+d_1+d_2+\ldots$ be an odd endomorphism of $E \otimes \Omega(B)$. Such that $d_0$ $E^0 \to E^1$ is a linear map, $d_1 : E \to \Omega^1(E)$ is a connection, and then $d_i$ for $i>1$ are bundle maps $E \to \Omega^i(E)$. Then the flatness condition $A^2=0$, does not force $d_1$ to be a flat connection. However, it does force $(d_0)^2=0$ which turns the fibers into chain complexes, and $d_1$ is compatible with $d_0$. Thus the induced (Gauss-Manin) connection $D=[d_1]$ on the cohomology $H(d_0,E)$ is flat.
See for example Abad-Schatz "Reidemeister torsion for flat superconnections" [https://arxiv.org/pdf/1108.5103.pdf]
