Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry of the Scaled Simplex," we prove that modular sweep maps are bijective. We conclude that the general sweep maps defined by D. Armstrong, N. Loehr, and G. Warrington in "Sweep Maps: A Continuous Family of Sorting Algorithms" are bijective. As a special case, this proves that the zeta map on rational Dyck paths is a bijection.