Topological algebra of symplectic geometry of symmetric powers

To a noncompact orientable surface with no closed boundary, we associate the sum of Fukaya categories of (Liouville sectors associated to) its symmetric powers. We construct sectorial covers with the combinatorics of the bar resolution to show this association extends to an open 2d topological field theory – without naming a Lagrangian, let alone a holomorphic disk. In particular, we recover results of Rouquier and Manion on extending Heegaard-Floer theory down to an interval.