Higher local systems and the categorified monodromy equivalence

Abstract

We study local systems of $(\infty,n)$-categories on spaces. We prove that categorical local systems are captured by (higher) monodromy data: in particular, if $X$ is $(n+1)$-connected, then local systems of $(\infty,n)$-categories over $X$ can be described as $\mathbb{E}_{n+1}$-modules over the iterated loop space $\Omega_{n+1}X$. This generalizes the classical monodromy equivalence presenting ordinary local systems as modules over the based loop spaces. Along the way we revisit from the perspective of $\infty$-categories Teleman's influential theory of topological group actions on categories, and we extend it to topological actions on $(\infty,n)$-categories. Finally, we show that the group of invertible objects in the category of local systems of $(\infty,n)$-categories over an $n$-connected space $X$ is isomorphic to the group of characters of $\pi_n(X)$. This should be thought of as a topological analogue of the higher Brauer group of the space $X$. We conclude the paper with applications of the theory of categorical local systems to the fiberwise Fukaya category of symplectic fibrations.