Fukaya category
In symplectic topology, a Fukaya category of a symplectic manifoldwhose objects are Lagrangian submanifolds of, and morphisms are Lagrangian Floer chain groups:Its finer structure can be described as an A∞-category.They are named after Kenji Fukaya who introduced thelanguage 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.For each pair of Lagrangian submanifoldsthat intersect transversely, one defines the Floer cochain complexwhich is a module generated by intersection pointsThe Floer cochain complex is viewed as the set of morphisms fromcategory, meaning that besides ordinary compositions, there are higher composition maps It is defined as follows.Choose a compatible almost complex structureof the cochain complexes, the moduli space ofhas a count in the coefficient ring.The sequence of higher compositionsrelations because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.This definition of Fukaya category for a general (compact) symplectic manifold has never been rigorously given.The main challenge is the transversality issue, which is essential in defining the counting of holomorphic disks.