Fukaya category

In symplectic topology, a Fukaya category of a symplectic manifold $$(X, \omega)$$ is a category $$\mathcal F (X)$$ whose objects are Lagrangian submanifolds of $$X$$, and morphisms are Lagrangian Floer chain groups: $$\mathrm{Hom} (L_0, L_1) = CF (L_0,L_1)$$. Its finer structure can be described as an A∞-category.

They are named after Kenji Fukaya who introduced the $$A_\infty$$ 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 now been computationally verified for a number of examples.

Formal definition
Let $$ (X, \omega) $$ be a symplectic manifold. For each pair of Lagrangian submanifolds $$ L_0, L_1 \subset X $$ that intersect transversely, one defines the Floer cochain complex $$ CF^*(L_0, L_1) $$ which is a module generated by intersection points $$ L_0 \cap L_1 $$. The Floer cochain complex is viewed as the set of morphisms from $$ L_0 $$ to $$ L_1 $$. The Fukaya category is an $$ A_\infty $$ category, meaning that besides ordinary compositions, there are higher composition maps


 * $$ \mu_d: CF^* (L_{d-1}, L_d) \otimes CF^* (L_{d-2}, L_{d-1})\otimes \cdots \otimes CF^*( L_1, L_2) \otimes CF^* (L_0, L_1) \to CF^* ( L_0, L_d). $$

It is defined as follows. Choose a compatible almost complex structure $$ J $$ on the symplectic manifold $$ (X, \omega) $$. For generators $$ p_{d-1, d} \in CF^*(L_{d-1},L_d), \ldots, p_{0, 1} \in CF^*(L_0,L_1) $$ and $$ q_{0, d} \in CF^*(L_0,L_d) $$ of the cochain complexes, the moduli space of $$ J $$-holomorphic polygons with $$ d+ 1 $$ faces with each face mapped into $$ L_0, L_1, \ldots, L_d $$ has a count


 * $$ n(p_{d-1, d}, \ldots, p_{0, 1}; q_{0, d}) $$

in the coefficient ring. Then define


 * $$ \mu_d ( p_{d-1, d}, \ldots, p_{0, 1} ) = \sum_{q_{0, d} \in L_0 \cap L_d} n(p_{d-1, d}, \ldots, p_{0, 1}) \cdot q_{0, d} \in CF^*(L_0, L_d)$$

and extend $$ \mu_d $$ in a multilinear way.

The sequence of higher compositions $$ \mu_1, \mu_2, \ldots, $$ satisfy the $$ A_\infty $$ relations 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.