# Infinity local systems

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

• Perhaps a more concrete model for the situation you're interested in is just (dg-)modules over $C_*(\Omega X)$? It's noncanonical since it involves choosing a basepoint, but it's at least concrete. Taking chains on the simplicial object $B_{\bullet}\Omega X$ gives a resolution of $k$ over $C_*(\Omega X)$ which yields the computation $hom(k_X, k_X)=C^*(B\Omega X)=C^*(X)$, for example. – Dylan Wilson Apr 6 at 18:14
• For question (2), the point is that $C^*(X;k)$ is the homotopy limit of $k_X$ seen as the constant functor from $X$ to (dg) $k$-modules, therefore the computation you want is just the universal property of the limit. – Denis Nardin Apr 6 at 18:24
• Look at proposition 5.3. in the following article www3.nd.edu/~wgd/Dvi/Duality.Algebra.Topology.pdf I guess you do need to assume more finiteness condition on $X$. – GSM Apr 6 at 18:30
• @GSM that proposition is referring to something else (though related). You do not need any finiteness assumptions for the statement I gave, or for what Denis said. – Dylan Wilson Apr 6 at 19:08
• @user142700 Local systems "really" means the functor category $\mathsf{Fun}(X, \mathsf{Ch})$ where one must interpret the space $X$ as yielding an $\infty$-category, and $\mathsf{Ch}$, likewise, as an $\infty$-category. In this setting, $k_X$ means the constant diagram $X \to \bullet \to \mathsf{Ch}$ at the chain complex $k$. The `tensor hom' adjunction tells you that $hom(k_X, -)$ is right adjoint to the functor assigning to a chain complex $C_*$ the constant diagram at $C_*$. On the other hand, the right adjoint to the constant diagram functor is called 'the (homotopy) limit'. – Dylan Wilson Apr 6 at 19:56