Fukaya category

Definition

This gives the simplest version, over the field $\F_2$, in the exact setting. Let $(M, \omega)$ be an exact symplectic manifold, $\omega = d\lambda$, that is convex at infinity. Fix a generic almost complex structure $J$ compatible with $\omega$ and of contact type at infinity.

1. The objects are the closed exact Lagrangian submanifolds $L \subset M$, those for which $\lambda|_L$ is exact.
2. Let $L_0, L_1$ be objects meeting transversally. The morphism space $\hom(L_0, L_1)$ is the $\F_2$-vector space $CF(L_0, L_1)$ with basis the finite set $L_0 \cap L_1$. For non-transverse pairs one first moves $L_1$ by a chosen Hamiltonian perturbation.
3. For $k \ge 1$ and objects $L_0, \ldots, L_k$, there is a map $$\mu^k \colon \hom(L_{k-1}, L_k) \otimes \cdots \otimes \hom(L_0, L_1) \to \hom(L_0, L_k)$$ It sends $x_k \otimes \cdots \otimes x_1$ to $\sum_y m(y; x_1, \ldots, x_k)\, y$. The number $m(y; x_1, \ldots, x_k) \in \F_2$ counts modulo $2$ certain pseudoholomorphic maps $u$ from a disc with $k + 1$ boundary punctures to $M$. These maps send the boundary arcs in cyclic order to $L_0, L_1, \ldots, L_k$ and converge to $x_1, \ldots, x_k$ and $y$ at the punctures. Only maps in moduli spaces of dimension $0$ are counted.

The Fukaya category of $M$ is this collection of objects, morphism spaces and maps $\mu^k$.