Fukaya category

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold ( M , ω ) {displaystyle (M,omega )} is a category F ( M ) {displaystyle {mathcal {F}}(M)} whose objects are Lagrangian submanifolds of M {displaystyle M} , and morphisms are Floer chain groups: H o m ( L 0 , L 1 ) = F C ( L 0 , L 1 ) {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. In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold ( M , ω ) {displaystyle (M,omega )} is a category F ( M ) {displaystyle {mathcal {F}}(M)} whose objects are Lagrangian submanifolds of M {displaystyle M} , and morphisms are Floer chain groups: H o m ( L 0 , L 1 ) = F C ( L 0 , L 1 ) {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 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.