Higher local systems and the categorified monodromy equivalence

We study local systems of $(\mathrm{\infty },n)$-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 $(\mathrm{\infty },n)$-categories over $X$ can be described as ${\mathbb{E}}_{n+1}$-modules over the iterated loop space ${\mathrm{\Omega }}_{n+1}X$. 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 $\mathrm{\infty }$-categories Teleman’s influential theory of topological group actions on categories, and we extend it to topological actions on $(\mathrm{\infty },n)$-categories. Finally, we show that the group of invertible objects in the category of local systems of $(\mathrm{\infty },n)$-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.