The symplectic (A-infinity,2)-category Symp, which is currently under construction by myself and my collaborators, is a 2-category-like structure whose objects are symplectic manifolds and where hom(M,N) := Fuk(M^- x N). Symp is a coherent algebraic structure which encodes the functoriality properties of the Fukaya category. This talk will begin with the following question: what can we say about the part of Symp that knows only about a single symplectic manifold M, and the diagonal Lagrangian correspondence from M to itself? We expect that the answer to this question should be a chain-level algebraic structure on symplectic cohomology, and in this talk I will present progress toward confirming this. Specifically, I will present a "simplicial version" of the 2-dimensional Fulton-MacPherson operad, which may be of independent topological interest. If there is time, I will also explain how this development can be used to give a definition of (A-infinity,2)-categories that involves only finitely many operations of each arity.
