The Topological Representation Theorem for matroids states that every matroid can be realized as an arrangement of codimension-one homotopy spheres on a sphere. Anderson and Engstrom, independently, showed how to explicitly construct such an arrangement for any given matroid. I will show that the structure-preserving maps between matroids induce topological mappings between their representations using Engstrom's construction. Specifically, I will show that weak maps induce continuous, (Z/2Z)-equivariant maps which weakly decrease Betti numbers. If time permits, I will also discuss how this process yields a functor from the category of matroids with weak maps to the homotopy category of topological spaces.