Higher operad structure for Fukaya categories
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912954717306880 |
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| author | Yuan, Hang |
| author_facet | Yuan, Hang |
| contents | Operads often arise from geometry. The standard $A_\infty$ operad can be derived from the cellular chains on the Stasheff associahedra, and an $A_\infty$ algebra is an algebra over this operad. The notion of an $\mathbf{fc}$-multicategory, also called a virtual double category, is a two-dimensional generalization of operads and multicategories. Here $\mathbf{fc}$ stands for the free category monad.
We establish a natural $\mathbf{fc}$-multicategory structure on the collection of moduli spaces of pseudo-holomorphic polygons with boundary on sequences of Lagrangian submanifolds in a symplectic manifold. These moduli spaces are known to underlie the construction of Fukaya categories. Based on this, we develop the theory of differential graded (dg) variants of $\mathbf{fc}$-multicategories and show that a broad range of $A_\infty$-type structures, such as $A_\infty$ algebras, $A_\infty$ (bi)modules, and $A_\infty$ categories (possibly curved), admit a uniform operadic formulation as algebras over dg $\mathbf{fc}$-multicategories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher operad structure for Fukaya categories Yuan, Hang Algebraic Topology Category Theory Quantum Algebra Symplectic Geometry Operads often arise from geometry. The standard $A_\infty$ operad can be derived from the cellular chains on the Stasheff associahedra, and an $A_\infty$ algebra is an algebra over this operad. The notion of an $\mathbf{fc}$-multicategory, also called a virtual double category, is a two-dimensional generalization of operads and multicategories. Here $\mathbf{fc}$ stands for the free category monad. We establish a natural $\mathbf{fc}$-multicategory structure on the collection of moduli spaces of pseudo-holomorphic polygons with boundary on sequences of Lagrangian submanifolds in a symplectic manifold. These moduli spaces are known to underlie the construction of Fukaya categories. Based on this, we develop the theory of differential graded (dg) variants of $\mathbf{fc}$-multicategories and show that a broad range of $A_\infty$-type structures, such as $A_\infty$ algebras, $A_\infty$ (bi)modules, and $A_\infty$ categories (possibly curved), admit a uniform operadic formulation as algebras over dg $\mathbf{fc}$-multicategories. |
| title | Higher operad structure for Fukaya categories |
| topic | Algebraic Topology Category Theory Quantum Algebra Symplectic Geometry |
| url | https://arxiv.org/abs/2603.08039 |