Functoriality in categorical symplectic geometry

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim-Woodward's quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya's alternate approach to defining functors between Fukaya $A_\infty$-categories, and the second author's ongoing construction of the symplectic $(A_\infty,2)$-category. In the last section, we describe a number of direct and indirect applications of this circle of ideas, and propose a conjectural version of the Barr-Beck Monadicity Criterion in the context of the Fukaya $A_\infty$-category.