Bordered Floer Homology and Lefschetz fibrations with corners

Lipshitz, Ozsv\'ath and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to each such cobordism between two mapping cylinders f and g. Like the Ozsv\'ath-Szab\'o invariants of cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the morphism F only depends on the symplectic structure of the cobordism in question.