I will report on joint work-in-progress with Satyan Devadoss, in which we aim to define a family of polytopes P_{nd,...,n1} called "2-associahedra", where (nd,...,n1) is a sequence of nonnegative integers with d>0 and nd+...+n1>0. This family specializes to the associahedra in two ways, as P_n and P_{0,...,0,1,0,...,0}; it specializes to the multiplihedra as P_{n,0}. These polytopes originate in symplectic geometry: they are expected to be the domain moduli spaces of certain pseudoholomorphic objects which will yield the right notion of functoriality for the Fukaya category, an invariant of symplectic manifolds.