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?