  We demonstrate an isomorphism between the homology of the strand algebra of
bordered Floer homology, and the category algebra of the contact category
introduced by Honda. This isomorphism provides a direct correspondence between
various notions of Floer homology and arc diagrams, on the one hand, and
contact geometry and topology on the other. In particular, arc diagrams
correspond to quadrangulated surfaces, idempotents correspond to certain basic
dividing sets, strand diagrams correspond to contact structures, and
multiplication of strand diagrams corresponds to stacking of contact
structures. The contact structures considered are cubulated, and the cubes are
shown to behave equivalently to local fragments of strand diagrams.
