Lagrangian shadows and triangulated categories

We introduce new metrics on spaces of Lagrangian submanifolds, not necessarily in a fixed Hamiltonian isotopy class. Our metrics arise from measurements involving Lagrangian cobordisms. We also show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some forms of rigidity of Lagrangian intersections. We also fit these constructions in the more general algebraic setting of triangulated categories, independent of Lagrangian cobordism. As a main technical tool, we develop aspects of the theory of (weakly) filtered A∞-categories.