---
tag: 4HS60JABC5
kind: concept
name: Fukaya category
msc: [53D37, 53D40, 18G70]
sources:
  - url: https://en.wikipedia.org/wiki/Fukaya_category
same_as:
  wikidata: Q5507585
  wikipedia: Fukaya category
---

## Summary

The 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.
Its composition laws count [[pseudoholomorphic|N469E47V75]] 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|B5N9WJH5F0]].

## Explanations

### L1 human

Lagrangian submanifolds are the most important subspaces of a symplectic manifold, for example curves on a surface.
Two of them, if in general position, meet in finitely many points, and these points span the space of arrows from one to the other.
To compose arrows, one counts holomorphic triangles whose three sides lie on three Lagrangians and whose corners are the given intersection points.
On 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.