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In symplectic topology, a Fukaya category of a symplectic manifold {\displaystyle } is a category F {\displaystyle {\mathcal {F}}} whose objects are Lagrangian submanifolds of M {\displaystyle M} , and morphisms are Floer chain groups: H o m = F C {\displaystyle \mathrm {Hom} =FC}. Its finer structure can be described in the language of quasi categories as an A∞-category.
They are named after Kenji Fukaya who introduced the A ∞ {\displaystyle A_{\infty }} language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are A∞-categories, they have associated derived categories, which are the subject of the celebrated homological mirror symmetry conjecture of Maxim Kontsevich. This conjecture has been computationally verified for a number of comparatively simple examples.