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 the celebrated homological mirror symmetry conjecture of Maxim Kontsevich. This conjecture has been computationally verified for a number of comparatively simple 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"?'

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