Several well known polytopal constructions are examined from the functorial point of view. A
naive analogy between the Billera–Sturmfels fiber polytope and the abelian kernel is …

If we start with defining affine hull, and then proving aff (A)= A,(since A is affine, and affine
hull is the smallest affine set containing A, so it contains and is equal to itself) then we have …