  Anomalous Floquet topological phases are a hallmark, without a static analog,
of periodically driven systems. Recently, Quantum Floquet Engineering has
emerged as an interesting approach to cavity-QED materials, which recovers the
physics of Floquet engineering in its semi-classical limit. However, the
mapping between these two widely different scenarios remains mysterious in many
aspects. We discuss the emergence of anomalous topological phases in cavity-QED
materials, and link topological phase transitions in the many-body spectrum
with those in the $0$- and $\pi$-gaps of Floquet quasienergies. Our results
allow to establish the microscopic origin of an emergent discrete
time-translation symmetry in the matter sector, and link the physics of
isolated many-body systems with that of periodically driven ones. Finally, the
relation between many-body and Floquet topological invariants is discussed, as
well as the bulk-edge correspondence.
