Higher local systems and the categorified monodromy equivalence
James Pascaleff, Emanuele Pavia, Nicolò Sibilla
TL;DR
The paper advances a categorified monodromy program: local systems of (∞,n)-categories over spaces are fully controlled by monodromy data encoded through iterated loop spaces, generalizing the classical monodromy of vector space local systems. It develops a unified framework translating categorical local systems into modules over loop-space algebras, extends Teleman’s topological action picture to ∞-categories and higher n-categories, and provides 2- and n-categorical enhancements of these equivalences. Under suitable connectivity hypotheses, the authors prove natural, functorial equivalences between LocSysCat^n(X) and iterated module categories over Ω_*^n X, describe how topological group actions correspond to actions on Hochschild-type invariants, and extend these ideas to higher Hochschild cohomology. The theory yields applications to the fiberwise Fukaya category in symplectic geometry, tying local systems of higher categories to geometric invariants and to Seidel-type phenomena via the categorified monodromy formalism. Altogether, this work furnishes a robust higher-categorical monodromy framework with substantial implications for symplectic geometry and higher Brauer-type structures.
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 $Ω_{n+1}X$. 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 $π_n(X)$. 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.