TITLE: Is the Fukaya category "defined"?
QUESTION [23 upvotes]: 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?

REPLY [10 votes]: Mike has given a good answer (basically - bubbling), but perhaps I can elaborate/add.
In general, due to disk bubbling one gets a curved or obstructed A-infinity category. The reason that's bad is that in this case m1 does not square to zero - no Floer homology. In the case when m0 is a multiple of fundamental class, then for CF(L,L) the A-infinity relations still say m1 squares to zero, and more generally for L_1 and L_2 if it's the same multiple \lambda. So you get completely disjointed categories for each \lambda. This happens for fibers of toric symplectic manifolds, for example (and on their mirror Landau-Ginsburg model we get the categories of singularities D^b sing, \lambda corresponding to the value of the superpotential). As Mike said, the business of bounding cochains is understanding when A-infinity relations can be modified to get m0 to be zero (unobstructed L) or multiple of fundamental class (weakly unobstructed L).
As for finding the "right" definition of Fukaya category, my understanding it is tricky business. Form the point of view of mirror symmetry, the current definition is a cheat, where we use passing to derived category to sweep under the rug all the problems - one of which is not dealing with immersed Lagrangians, and another is perhaps ignoring the suggestion of Kapustin-Orlov to include coisotropics. As I understand people are studying ways to do Floer theory for both of these new types of objects. This is however, a somewhat different question from the one you asked.