The set-theoretic diagonal of a polytope has the crippling defect of not being cellular: its image is not a union of cells. One is thus looking for a cellular approximation to the diagonal. Finding such an approximation in the case of the simplices and the cubes is of fundamental importance in algebraic topology: it allows one to define the cup product in cohomology. I will present a general method, coming from the theory of fiber polytopes of Billera and Sturmfels, which permits to solve this problem for any family of polytopes. I will then sketch how this machinery, applied to new families of polytopes, gives us the tools to define higher algebraic objects such as the tensor product of homotopy operads or a functorial tensor product of A-infinity categories.