  Time-periodic driving of a quantum system can enable new dynamical
topological phases of matter that could not exist in thermal equilibrium. We
investigate two related classes of dynamical topological phenomena in 2D
systems: Floquet symmetry protected topological phases (FSPTs), and Floquet
enriched topological orders (FETs). By constructing solvable lattice models for
a complete set of 2D bosonic FSPT phases, we show that bosonic FSPTs can be
understood as topological pumps which deposit loops of 1D SPT chains onto the
boundary during each driving cycle, which protects a non-trivial edge state by
dynamically tuning the edge to a self-dual point poised between the 1D SPT and
trivial phases of the edge. By coupling these FSPT models to dynamical gauge
fields, we construct solvable models of FET orders in which anyon excitations
are dynamically transmuted into topologically distinct anyon types during each
driving period. These bosonic FSPT and gauged FSPT models are classified by
group cohomology methods. In addition, we also construct examples of "beyond
cohomology" FET orders, which can be viewed as topological pumps of 1D
topological chains formed of emergent anyonic quasi-particles.
