TITLE: Infinity local systems
QUESTION [13 upvotes]: I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems". 
From what I've been told, given a good cover $\{U_i\}$ of $X$, an infinity local system on a connected space $X$ assigns:
-a chain complex $C_i$ to every contractible open set $U_i$
-a chain morphism $F_{ij}: C_i \to C_j$ to every double intersection $U_i \cap U_j$ 
-a homotopy of chain morphisms $F_{ik} \sim F_{jk} \circ F_{ij}$ for every triple intersection $U_i \cap U_j \cap U_k$ 
-etc.
EDIT: actually, I think that the above definition is rather an "infinity-sheaf". An infinity local system should be a locally constant "infinity-sheaf". My guess is that the Maurer-Cartan condition mentioned below is precisely encoding the "locally-constant" condition.
Some questions:
(1) What are some good sources to learn about these objects (from the perspective of a second year graduate student with limited exposure to these ideas)? 
The only paper I have found is Section 2.1 of "A Riemann–Hilbert correspondence for infinity local systems"  (https://arxiv.org/pdf/0908.2843.pdf)  In this paper, an infinity local system is defined as a set of maps from a simplicial set $K$ to a differential graded category $C$ which satisfy a certain Maurer-Cartan equation. However, it's not clear to me how to make sense of this definition (e.g. what is the role of Maurer-Cartan? how does this definition match up with the intuitive notion of an infinity local system described above?)
(2) The infinity local systems on a topological space $X$ are supposed to form a dg category. I recently heard a talk in which the speaker claimed (as if it was the most natural thing in the world:) )that $hom(k_X, k_X)=C^*(X;k)$, where $k$ is some field, $k_X$ is the locally constant sheaf with stalk $k$ is degree $0$ and $C^*(X;k)$ is the singular cohomology of $X$ with $k$ coefficients. 
Is there a good way to understand why $hom(k_X, k_X)=C^*(X;k)$? More generally, how should one understand the dg category structure on $\operatorname{Loc}_{\infty}(X)$?

REPLY [3 votes]: Since you tagged this with "symplectic geometry", I'm going to give an answer from a symplectic geometry perspective, which may not be what you're looking for, but (as a symplectic person) I find it is a helpful point of view. This will use the language of $A_\infty$-categories as well as dg-categories.
Given a manifold $X$, take its cotangent bundle $T^*X$. This is a non-compact symplectic manifold. You can consider its Fukaya category of exact embedded Lagrangian submanifolds which agree with the symplectisation of a Legendrian at infinity. The Floer cochain groups (morphisms) between $L_1,L_2$ are free k-vector spaces on intersection points between $L_1$ and $\phi(L_2)$, where $\phi$ is the time 1 map of a suitable Hamiltonian. You have to specify a suitable class of Hamiltonians, and I want to use Hamiltonians which are "quadratic at infinity", in other words they look something like kinetic energy with respect to some Riemannian metric. Since kinetic energy as a Hamiltonian generates geodesic flow, the Floer cochains are something like (k-linear combinations of) geodesics connecting the Lagrangians.
For example, if $L_1$ and $L_2$ are cotangent fibres at $x_1$ and $x_2$ then the Floer complex is something like the free k-vector space on the set of geodesics from $x_1$ to $x_2$. It is then a theorem (of Abbondandolo and Schwartz at the level of homology beefed up to the $A_\infty$ level by Abouzaid) that the Floer complex between two cotangent fibres is quasiisomorphic to chains on the space of paths between $x_1$ and $x_2$ (and to $C_{-*}(\Omega X)$ as an $A_\infty$-algebra when $x_1=x_2$ and concatenation makes sense).
Abouzaid also showed that a cotangent fibres generates this Fukaya category, so you get a fully faithful Yoneda functor from the Fukaya category to the dg-category of $A_\infty$-bimodules over chains on the based loop space. In other words, if you want to compute the Floer complex between two Lagrangians $L_1$ and $L_2$ and you have a cotangent fibre $F$, you can take the $CF(F,F)$-bimodules $CF(F,L_n)\otimes CF(L_n,F)$, $n=1,2$, and take homs between them in the category of $A_\infty$ $CF(F,F)$-bimodules. Since $CF(F,F)\cong C_{-*}(\Omega X)$, this is quasiequivalent to the category of infinity local systems. So this category of infinity local systems is the Fukaya category of $T^*X$.
Now how do you see that $hom(k,k)=C^*(X)$? Well, there is a Lagrangian in $T^*X$ whose Floer complex is $C^*(X)$, namely the zero section. For example, a small Hamiltonian deformation of the zero section is a graph of an exact 1-form $df$, and the intersection points between this and the zero section happen at critical points of $f$; in fact Floer showed that in this case the Floer complex is the Morse complex for a suitable choice of almost complex structures.
What is the Yoneda bimodule corresponding to the zero section? Well the zero section intersects our cotangent fibre at a single point, so $CF$ is just k (our field, considered as a trivial $A_\infty$-module over $C_{-*}(\Omega X)$). Its self homs in the category of $C_{-*}(\Omega X)$-bimodules should therefore compute $C^*(X)$.
The relevant papers of Abouzaid are:
https://arxiv.org/abs/0907.5606
https://arxiv.org/abs/1003.4449