  We develop the 2-representation theory of the odd one-dimensional super Lie
algebra $gl(1|1)^+$ and show it controls the Heegaard-Floer theory of surfaces
of Lipshitz, Ozsv\'ath and Thurston. Our main tool is the construction of a
tensor product for 2-representations. We show it corresponds to a gluing
operation for surfaces, or the chord diagrams of arc decompositions. This
provides an extension of Heegaard-Floer theory to dimension one, expanding the
work of Douglas, Lipshitz and Manolescu.
