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  "body": "\n## Summary\n\nThe Fukaya category of a symplectic manifold is an algebraic structure whose objects are [[Lagrangian submanifolds|H8ACPTMX08]] and whose morphisms are Lagrangian [[Floer|NCKFPAGHXF]] cochain complexes.\nIts composition laws count [[pseudoholomorphic|N469E47V75]] polygons with boundary on the Lagrangians.\nComposition is associative only up to homotopy, so it is an $A_\\infty$-category rather than an ordinary category.\nIt was introduced by Kenji Fukaya and is one side of [[homological mirror symmetry|B5N9WJH5F0]].\n\n## Explanations\n\n### L1 human\n\nLagrangian submanifolds are the most important subspaces of a symplectic manifold, for example curves on a surface.\nTwo of them, if in general position, meet in finitely many points, and these points span the space of arrows from one to the other.\nTo compose arrows, one counts holomorphic triangles whose three sides lie on three Lagrangians and whose corners are the given intersection points.\nOn a torus, for example, closed curves of different slopes play the role of the Lagrangians, and the triangles are flat triangles in the plane that covers the torus.\n",
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