An equivariant, ample line bundle on a toric variety defines a polytope in a vector space. We extend this simple correspondence to a functor from the derived category of coherent sheaves on the toric variety to a category of constructible sheaves on the vector space. We prove this functor is an equivalence, thus categorifying Morelli's description of the equivariant K-theory of a toric variety. We connect this construction to mirror symmetry by two processes: T-duality and microlocalization. T-duality relates coherent sheaves on the toric variety to a Fukaya category on the cotangent bundle of the vector space. Microlocalization relates this Fukaya category to constructible sheaves on the vector space. This is a joint work with Bohan Fang, David Treumann, and Eric Zaslow.