TITLE: Representation theory of higher homotopy groups
QUESTION [7 upvotes]: I've seen some works on the representation of fundamental groups, which are (at least for me) quite important topic in mathematics. For example, Riemann-Hilbert correspondence relates representation of a fundamental group of complex algebraic variety with differential equations on it. (There's also a result in positive characteristic by Bhatt-Lurie).  Also, etale fundamental group and its representation is also important in number theory because it is closely related to Galois groups. (I don't know much about this though). However, I can't find any works that studies about representation of higher homotopy groups $\pi_{n}$ (or etale homotopy groups, whatever it is). Although $\pi_n$ is abelian for $n\geq 2$, there might exist some nontrivial stuff that $\pi_n$ can act naturally so that we can study about its representations (which would be 1-dimensional if irreducible, though). Could you give any examples if there's any interesting works about such things?

REPLY [11 votes]: There are many results that generalize the Riemann–Hilbert correspondence from the fundamental groupoid to the fundamental ∞-groupoid, for example:

Jonathan Block, Aaron Smith.  A Riemann–Hilbert correspondence for infinity local systems.

Joseph Chuang, Julian Holstein, Andrey Lazarev.  Maurer-Cartan moduli and theorems of Riemann-Hilbert type.

Manuel Rivera, Mahmoud Zeinalian.  The colimit of an ∞-local system as a twisted tensor product.

Camilo Arias Abad, Florian Schaetz.  Flat Z-graded connections and loop spaces.

Julian V. S. Holstein.  Morita cohomology.

Camilo Arias Abad, Florian Schaetz.  The A_infty de Rham theorem and integration of representations up to homotopy.

Vincent Braunack-Mayer.  Strict algebraic models for rational parametrised spectra II.

Camilo Arias Abad, Sebastian Velez Vasquez.  The higher Riemann-Hilbert correspondence and principal 2-bundles.


For the higher Galois theory, see

Marc Hoyois.  Higher Galois theory.