A stable infinity-category of Lagrangian cobordisms

Given an exact symplectic manifold M and a support Lagrangian Λ, 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 Λ. 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 Λ. 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.

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