Higher local systems and the categorified monodromy equivalence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912288675463168 |
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| author | Pascaleff, James Pavia, Emanuele Sibilla, Nicolò |
| author_facet | Pascaleff, James Pavia, Emanuele Sibilla, Nicolò |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher local systems and the categorified monodromy equivalence Pascaleff, James Pavia, Emanuele Sibilla, Nicolò Algebraic Topology Algebraic Geometry Category Theory K-Theory and Homology 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. |
| title | Higher local systems and the categorified monodromy equivalence |
| topic | Algebraic Topology Algebraic Geometry Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2501.10241 |