Fukaya category

Summary

The Fukaya category of a symplectic manifold is an algebraic structure whose objects are Lagrangian submanifolds and whose morphisms are Lagrangian Floer cochain complexes. Its composition laws count pseudoholomorphic polygons with boundary on the Lagrangians. Composition is associative only up to homotopy, so it is an $A_\infty$-category rather than an ordinary category. It was introduced by Kenji Fukaya and is one side of homological mirror symmetry.