The derived category of a variety is an important and difficult invariant. In this talk, I discuss a purely convex-geometric and combinatorial approach to these categories for toric varieties. Along the way, we will run into the curious question of studying the contractibility of set differences of polytopes.  I will focus on the toric variety of the permutahedron which has played an important role in many recent developments in matroid theory. I will discuss how this derived category can be described through the theory of matroids.