# Fukaya category

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold ${\displaystyle (M,\omega )}$ is a category ${\displaystyle {\mathcal {F}}(M)}$ whose objects are Lagrangian submanifolds of ${\displaystyle M}$, and morphisms are Floer chain groups: ${\displaystyle \mathrm {Hom} (L_{0},L_{1})=FC(L_{0},L_{1})}$. 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 ${\displaystyle A_{\infty }}$ 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.