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.

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

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

I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".

From what I've been told, 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.

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

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.

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