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  "body": "\n## Definition\n\nThis gives the simplest version, over the field $\\F_2$, in the exact setting.\nLet $(M, \\omega)$ be an exact [[symplectic manifold|CAN490EPRT]], $\\omega = d\\lambda$, that is convex at infinity.\nFix a generic [[almost complex structure|9SYWRGVTRS]] $J$ compatible with $\\omega$ and of contact type at infinity.\n\n1. The objects are the closed exact [[Lagrangian submanifolds|H8ACPTMX08]] $L \\subset M$, those for which $\\lambda|_L$ is exact.\n2. 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.\n3. For $k \\ge 1$ and objects $L_0, \\ldots, L_k$, there is a map\n$$\\mu^k \\colon \\hom(L_{k-1}, L_k) \\otimes \\cdots \\otimes \\hom(L_0, L_1) \\to \\hom(L_0, L_k)$$\nIt 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|N469E47V75]] 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.\n\nThe *Fukaya category* of $M$ is this collection of objects, morphism spaces and maps $\\mu^k$.\n\n## Remarks\n\nThe maps satisfy the $A_\\infty$ relations: for every $k \\ge 1$,\n$$\\sum_{m + n \\le k} \\mu^{k - m + 1}\\left(x_k, \\ldots, x_{n + m + 1}, \\mu^m(x_{n + m}, \\ldots, x_{n + 1}), x_n, \\ldots, x_1\\right) = 0,$$\nwith $m \\ge 1$ and $n \\ge 0$ (over $\\F_2$ there are no signs).\nFor $k = 1$ this says $\\mu^1 \\circ \\mu^1 = 0$, and $\\mu^1$ is the Lagrangian [[Floer|NCKFPAGHXF]] differential.\nFor $k = 2$ it says that $\\mu^2$ is a chain map, and for $k = 3$ that $\\mu^2$ is associative up to the homotopy $\\mu^3$.\nSo the cohomology groups $HF(L_0, L_1)$ with the product induced by $\\mu^2$ form an ordinary category, the Donaldson–Fukaya category.\n\nExample: in $T^*S^1 = \\R \\times S^1$ with $\\lambda = p\\,d\\theta$, the zero section and the graph of $d f$ for a function $f$ on $S^1$ are exact Lagrangians.\nA non-example of an object is a circle $\\set{p = c}$ with $c \\ne 0$: there $\\lambda|_L = c\\, d\\theta$ is closed but not exact.\n\nThe construction for closed or non-exact manifolds needs much more work, because of disc bubbling; Fukaya, Oh, Ohta and Ono developed it with obstruction theory.\nWorking over $\\Z$ or with a $\\Z$-grading needs extra data on the Lagrangians, such as spin structures and gradings.\n",
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