equivalences in/of $(\infty,1)$-categories
The Fukaya category of a symplectic manifold $X$ is an A-∞ category having Lagrangian submanifolds of $X$ as objects. When two Lagrangian submanifolds $L_1$ and $L_2$ of $X$ meet transversally, their hom-space in the Fukaya category can be roughly defined as the free vector space generated by the intersection points $x\in L_1\cap L_2$; one of the main difficulties in giving a rigorous definition of the Fukaya category in general relies precisely in the problem of correctly defining the hom-spaces for nontransversal intersections. As one could expect, the same difficulty carries on to the definition of the multilinear operations in the Fukaya category: when Lagrangians $L_1, L_2,\dots,L_{k+1}$ intersect transversally one has a clear geometric intuition of the multiplication
in terms of counting pseudo-holomorphic disks into $X$ whose boundaries lie on the given Lagrangian submanifolds, but when intersections are nontransverse, the definition of $m_k$ becomes more evasive.
In string theory, the Fukaya category of a symplectic manifold $X$ represents the category of D-branes in the A-model – the A-branes – with target space $X$. For Landau-Ginzburg models, the category of A-branes is described by Fukaya-Seidel categories.
The assignment that sends a symplectic manifold to its Fukaya category extends to a functor on a variant of the symplectic category with Lagrangian correspondences as morphisms. This is supposed to be the FQFT incarnation of Donaldson theory. See at Lagrangian correspondences and category-valued TFT for more on this.
Fukaya categories have first been introduced in
The definitive reference is the book
See also
A beginners introduction is given by Denis Auroux, see link, a text (available as ArXiv 1301.7056), based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Université de Nantes in June 2011.
Discussion of the relation to Lagrangian cobordism is in