Fukaya category

In symplectic topology, a discipline within mathematics, a Fukaya category of a symplectic manifold $(M, \omega)$ is a category $\mathcal F (M)$ whose objects are Lagrangian submanifolds of $M$, and morphisms are Floer chain groups: $\mathrm{Hom} (L_0, L_1) = FC (L_0,L_1)$. Its finer structure can be described in the language of quasi categories as an A_∞-category.

They are named after Kenji Fukaya who introduced the $A_\infty$ language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are A_∞-categories, they have associated derived categories, which are the subject of a celebrated conjecture of Maxim Kontsevich: the homological mirror symmetry. This conjecture has been verified by computations for a variety of comparatively simple examples.(Examples?)

References

• P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich lectures in Advanced Mathematics
• Fukaya, Y-G. Oh, H. Ohta, K. Ono, Lagrangian Intersection Floer Theory, Studies in Advanced Mathematics
• The thread on MathOverflow 'Is the Fukaya category "defined"?'