Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold ${\displaystyle (X,\omega )}$ is a category ${\displaystyle {\mathcal {F}}(X)}$ whose objects are Lagrangian submanifolds of ${\displaystyle X}$, and morphisms are Lagrangian Floer chain groups: ${\displaystyle \mathrm {Hom} (L_{0},L_{1})=CF(L_{0},L_{1})}$. Its finer structure can be described as an A_∞-category.

They are named after Kenji Fukaya who introduced the ${\displaystyle A_{\infty }}$ language first in the context of Morse homology,^[1] 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.^[2] This conjecture has now been computationally verified for a number of examples.