A stable ∞-category of Lagrangian cobordisms

Abstract

Given an exact symplectic manifold M and a support Lagrangian Λ⊂M, we construct an ∞-category LagΛ(M) 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 LagΛ(M) are Lagrangian branes inside of M×T⁎Rn, 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 LagΛ(M) is a stable ∞-category, and in particular its homotopy category is triangulated, with mapping cones given by an elementary construction. The shift functor is equivalent to the familiar shift of grading for Lagrangian branes.