Sometimes people say that the Fukaya category is "not yet defined" in general.


What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact symplectic manifolds, not Fukaya-Seidel categories or wrapped Fukaya categories or whatever else might be out there)


What should a "correct" definition of the Fukaya category satisfy? (--- aside from, perhaps, "it makes homological mirror symmetry true")


What are some of the things which make defining the Fukaya category difficult?


What are some cases in which we "know" that we have the "correct" Fukaya category?



Is the Fukaya category "defined"?
From what I understand, the most fundamental issue obstructing the definition of the Fukaya category in general is the fact that the boundaries of the relevant moduli spaces typically have codimension-one pieces arising from the bubbling off of pseudoholomorphic discs. In situations where there are no bubbles (for instance, there isn't any bubbling when the Lagrangians are exact, by Stokes' theorem and the fact that the symplectic form would have to integrate positively over the bubble), the A-infinity relations are proven by interpreting the various terms as arising from boundary components of these moduli spaces--hence bubbling contributes additional terms which potentially mess the A-infinity relations up.


Fukaya-Oh-Ohta-Ono (at least in the version that I looked at a few years ago) develop an obstruction theory which assigns a sequence of classes in the homology of a given Lagrangian L (arising by evaluating boundaries of pseudoholomorphic discs) such that the obstruction classes of L all have to vanish in order for L to fit into the Fukaya category. And if the obstruction classes do vanish, one needs to choose a "bounding chain" b for L with the isomorphism type of (L,b) potentially depending on b.


There also some issues with defining the Fukaya category having to do with transversality of the moduli spaces, but my impression is that these issues are technical and can/have been addressed--and that the need to address them is part of what accounts for the length of the [current version](http://www.ams.org/bookstore?fn=20&arg1=amsipseries&ikey=AMSIP-46) of FOOO.


To learn more, you could do worse than spending some time at [Fukaya's webpage](http://www.math.kyoto-u.ac.jp/~fukaya/fukaya.html). In particular, the introduction to the old version of FOOO, which you'll find there, lays out some of the properties that the Fukaya category has and/or is supposed to have.


