A stable infinity-category of Lagrangian cobordisms
  Given an exact symplectic manifold M and a support Lagrangian \Lambda, we construct an infinity-category Lag, which we conjecture to be equivalent (after specialization of the coefficients) to the partially wrapped Fukaya category of M relative to \Lambda. Roughly speaking, the objects of Lag are Lagrangian branes inside of M x T*(R^n), for large n, and the morphisms are Lagrangian cobordisms that are non-characteristic with respect to \Lambda. The main theorem of this paper is that Lag is a stable infinity-category, so that its homotopy category is triangulated, with mapping cones given by an elementary construction. In particular, its shift functor is equivalent to the familiar shift of grading for Lagrangian branes. 
David Nadler and Hiro Lee Tanaka
math.SG