Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold formula_1 is a category formula_2 whose objects are Lagrangian submanifolds of formula_3, and morphisms are Floer chain groups: formula_4. 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 formula_5 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.
Formal definition.
Let formula_6 be a symplectic manifold. For each pair of Lagrangian submanifolds formula_7, suppose they intersect transversely, then define the Floer cochain complex formula_8 which is a module generated by intersection points formula_9. The Floer cochain complex is viewed as the set of morphisms from formula_10 to formula_11. The Fukaya category is an formula_12 category, meaning that besides ordinary compositions, there are higher composition maps
It is defined as follows. Choose a compatible almost complex structure formula_14 on the symplectic manifold formula_6. For generators formula_16 of the cochain complexes on the left, and any generator formula_17 of the cochain complex on the right, the moduli space of formula_14-holomorphic polygons with formula_19 faces with each face mapped into formula_20 has a count
in the coefficient ring. Then define
and extend formula_23 in a multilinear way.
The sequence of higher compositions formula_24 satisfy the formula_12 relation because the boundaries of various moduli spaces of holomorphic polygons correspond to configurations of degenerate polygons.