I am reading a paper by Block and Smith which is available here: https://www.sciencedirect.com/science/article/pii/S0001870813004118?via%3Dihub.
The category of infinity-local systems $\text{Loc}_\infty^C(K)$ is given where $C$ is the dg-category of (bounded) cochain complexes and $K$ a simplicial set. There is a filtration defined on the morphisms between two infinity-local systems by $F^k\text{Loc}_\infty^C(K)(F,G) = \{\phi \in \text{Loc}_\infty^C(K)(F,G) | \phi^i = 0 \text{ for } i < k \}$. From this we get an associated spectral sequence in which $E^{pq}_0 = \{\phi : K_p \to C^q\}$ with differential given pointwise by the dg-differential $d : C^q \to C^{q+1}$. In Corollary 2.10 it is claimed that the $E_1$ term of the spectral sequence is a local system valued in graded vector spaces in the ordinary sense.
I have trouble understanding this statement. I only know the notion of local systems in the context of sheaves on a topological space. I suspect that in this context it is meant to be $K = Sing_\infty(X)$- But even then I have no idea in which sense the $E_1$ is supposed to be a local system let alone a sheaf.
I would be very happy if someone has an idea what is going on here, especially if someone has read the paper.
This will be an incomplete answer: I can explain how they use the term ''local system'' here, but I have not looked enough into all the differentials and so on to know what exactly the local system associated to the $E_1$-page is (and I also find their (lack of) notation somewhat confusing).
They use the term ''local system'' for something more general than a sheaf: given a space $X$ and a category $\mathcal{C}$, with a ($\mathcal{C}$-valued) local system commonly is meant a functor $\Pi_1(X)\to\mathcal{C}$, where $\Pi_1(X)$ is the fundamental groupoid of $X$. In the same fashion, an $\infty$-local system on a space $X$ is just an $\infty$-functor $\Pi_\infty(X)\to\mathscr{C}$ for some $\infty$-category $\mathscr{C}$ (in your case the underlying $\infty$-category of a dg-category), and $\Pi_\infty(X)$ is the underlying homotopy type of the space $X$ (seen as $\infty$-groupoid). Now, for a simplicial set $K$, there is likewise a fundamental groupoid $\Pi_1(K)$, and then a local system is a functor $\Pi_1(K)\to\mathcal{C}$, and if $K$ is a Kan complex or more generally a quasicategory, then the definition the paper gives of an $\infty$-local system on $K$ is also the correct model of an $\infty$-functor $K_\infty\to \mathscr{C}$, where for clarity $K_\infty$ denotes the quasicategory $K$ seen as $\infty$-category. Note that, if $K$ is a Kan complex, $K_\infty$ ''is'' the associated homotopy type.
For simplicial sets there is another, related definition of a local system: if we let $\Delta(K)$ be the category of simplices of $K$ (so objects are the simplices in arbitrary degree of $K$, and morphisms are the structure maps between these simplices in $K$), then sometimes people mean with ''a local system on $K$'' a functor $\Delta(K)\to\mathcal{C}$. The two definitions for simplicial sets are closely related, because a functor $\Delta(K)\to\mathcal{C}$ that sends all morphisms in $\Delta(K)$ to isomorphisms (as you often do in local systems) factors uniquely through the localization $\Delta(K)[\text{all morphisms}^{-1}]\simeq\Pi_1(K)$, so the first definition of a local system is a special case of the second one.
In your case, what they mean is that the things you find in the $E_1$ page should give you a functor $\Delta(K)\to \mathrm{grVect}_k$, and based on their usage of Corollary 2.10 I suspect it will factor through the localization $\Delta(K)[\text{all morphisms}^{-1}]\simeq\Pi_1(K)$. What exactly this functor $\Delta(K)\to \mathrm{grVect}_k$ is, I don't know, as I said above. If it makes sense with all the differentials, maybe it can be an assignment that sends a $p$-simplex $\sigma$ of $K$ to the graded vector space whose $q$-th component consists of those elements of $E_1^{pq}$ represented by maps $\varphi\colon K_p\to \mathcal{C}^q$ in $E_0^{pq}$ sending all $p$-simplices except $\sigma$ to $0$.