Sutured TQFTs and Floer homology

The bordered Floer homology of Lipshitz, Ozsvath, and Thurston was interpreted by Auroux as defining an element in the partially wrapped Fukaya category of a symmetric product of the boundary. We naturally expect that this assignment should be functorial; e.g. a cobordism between two manifolds with torus boundary should induce a morphism between the corresponding Lagrangians. I’ll describe a framework for thinking about functoriality in terms of sutured manifolds and describe what it looks like for Heegaard Floer homology.