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.