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 $\operatorname{Loc}\_{\infty}(X)$? 



Infinity local systems
One way to discuss $\infty$-local systems over a space $X$ is in terms of the *fundamental $\infty$-groupoid* of $X$. To motivate this, recall that for classical local systems, one has the equivalence of categories
$$LocSys(X) \simeq Rep(\pi\_{1}(X)) $$
between the local systems on $X$ and representations of the fundamental group. It's easily verified that representations of the fundamental group are equivalent to functors out of the fundamental groupoid of $X$, so it's more natural (in the sense of not making any choices) to write
$$LocSys(X) \simeq Fun(\Pi\_{1}(X), Vec\_{k}) $$


The idea behind $\infty$-local systems is then to replace vector spaces with chain complexes, which we think of as a homotopical/derived version of vector spaces. The problem with just doing this naively though (say, by taking $Fun(\Pi\_{1}(X), Ch\_{k})$ where $Ch\_{k}$ is the category of chain complexes over $k$), is that the fundamental groupoid only encodes information about the 1-truncated homotopy type of $X$. If we're mapping into a category that carries higher homotopical data, there's no way we're going to get a satisfactory answer unless we encode the entire homotopy type of $X$ in the domain.


So instead, we replace $\Pi\_{1}(X)$ by $\Pi\_{\infty}(X)$ - which is probably more familiarly written as the singular simplicial set associated to $X$, denoted $Sing(X)$. Then, we can take as our definition of infinity local systems
$$LocSys^{\infty}(X) := Fun(\Pi\_{\infty}(X), Ch\_{k})$$
where we view $Ch\_{k}$ as an $\infty$-category. I'm not entirely sure how this definition fits in with that of the paper you cited, but it's certainly a common way to think of infinity local systems.


Some places to read:


1. **Appendix A of Higher Algebra.** This is definitely overkill for your question, as it mostly pertains to $\infty$-constructable sheaves, but it's a great place to learn some sheaf theory from the above perspective.
2. **Kerodon, Section 2.5**. This isn't directly about infinity local systems, but gives a lot of insight into the interplay between dg-categories and $\infty$-categories. (In particular, this process of 'taking chains' of a simplicial set mentioned in the comments by Dylan is explained here in detail).


As for your second question, the comments give a pretty concrete and conceptual answer as to how to perform this computation, but there's another heuristic that explains why you would expect this to be true without computing anything. The key point is that the dg-category $LocSys^{\infty}(X)$ is *a differential graded enhancement of the derived category of local systems on X.* This means that when we pass to the homotopy category, we recover the derived category of local systems.


Now in the (underived, 1-categorical) category of local systems, $hom(k\_{X}, \mathcal{F})$ is equivalent to just taking global sections of $\mathcal{F}$. So, when we pass to the derived category, $hom(k\_{X}, k\_{X})$ is just taking derived global sections of the constant sheaf - i.e. singular cohomology of $X$ with coefficients in $k$. Thus, in the dg-category $LocSys^{\infty}(X)$, the mapping complex $hom(k\_{X}, k\_{X})$ is some chain complex whose $i^{th}$ cohomology is isomorphic to $H^{i}(X ; k)$. So it shouldn't be surprising at all that this is $C^{\star}(X; k)$.


