Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C\_\*(\Omega X,k)$ of chains of loops on $X$. (I think this is equivalent, at least when $\text{char}(k)=0$, to locally constant sheaves where "sheaf" is defined in an appropriate higher-topos sense). 


I would like some construction of "$\infty$-constructible sheaves" which includes both this category and also constructible complexes of sheaves for some fixed stratification of $X$. Do topologists know of one? Is there an analogue of the Riemann-Hilbert correspondence for $X$ a complex algebraic manifold?



What is a higher derived constructible sheaf
In the appendix of "Higher Algebra" (<http://www.math.harvard.edu/~lurie/papers/HigherAlgebra.pdf>), Jacob Lurie describes constructible sheaves on a stratified space as representations of an exit path $\infty$-category. So one option is to take representations of this exit path $\infty$-category valued in the stable $\infty$-category (or appropriate DG category) of complexes of vector spaces. When the stratification is trivial, this recovers the usual notion of $\infty$-local system (as representations of the fundemental $\infty$-groupoid).


I would also be interested to hear about a Riemann-Hilbert Correspondence in this generality. In the local systems setting, there is Aaron Smith's thesis: <http://repository.upenn.edu/cgi/viewcontent.cgi?article=1462&context=edissertations>.


