What connects Symplectic Geometry so deeply with Algebraic Geometry and Low Dimensional Topology?

The answer that I will propose is a higher categorical structure in symplectic geometry that includes all Fukaya categories and geometric functors between them. Its highly involved algebraic and analytic details are the topic of joint work with Nate Bottman. However, its basic structure can be understood as a natural -- yet not previously studied -- extension of the category of topological spaces and continuous maps. So the goal of this talk is to make broadly accessible the two underlying ideas: The notion of quilts (collections of maps related by ``seam conditions'') and the string diagram approach to constructing and making sense of 2-categories.