Seminal work of Floer and Gromov in the late 1980s showed that the geometry of a symplectic manifold M can be probed by studying the pseudoholomorphic curves inside M. This approach has given rise to a variety of symplectic invariants, including the Fukaya category, which encodes an intersection theory of Lagrangian submanifolds..

The functorial properties of the Fukaya category are surprisingly subtle. Since 2010, work of Ma'u--Wehrheim--Woodward, Fukaya, and myself has led to a blueprint, partially proven, for a complete functoriality package, based on the notion of pseudoholomorphic quilts.

One version of this package is called the "symplectic (A-infinity,2)-category", or "Symp", for short. I will survey this package, following my recent paper with Mohammed Abouzaid (arXiv:2210.11159). I will then describe current work: first, work with Abouzaid and Poliakova that aims to equip Fukaya categories of Lagrangian torus fibrations with monoidal structures; and second, work with Abouzaid that aims to establish a version of the Barr--Beck theorem for the Fukaya category.