The Fukaya category pairs with Lagrangian cobordisms exactly
  We prove that a pairing between the Fukaya category and the oo-category of Lagrangian cobordisms respects mapping cones. This is another step toward constructing a lift of Fukaya categories to the level of spectra (in the sense of stable homotopy theory). As corollaries, we show that the map in our previous work from cobordism groups to Floer cohomology lifts to the level of spectra, and one also recovers some results of Biran and Cornea for what we call "vertically collared" cobordisms. 
Hiro Lee Tanaka
math.SG math.AT