  It is shown that the stroboscopic time-evolution under a Floquet unitary, in
any spatial dimension, and of any Hermitian operator, can be mapped to an
operator Krylov space which is identical to that generated by the edge operator
of the non-interacting Floquet transverse-field Ising model (TFIM) in
one-spatial dimension, and with inhomogeneous Ising and transverse field
couplings. The latter has four topological phases reflected by the absence
(topologically trivial) or presence (topologically non-trivial) of edge modes
at $0$ and/or $\pi$ quasi-energies. It is shown that the Floquet dynamics share
certain universal features characterized by how the Krylov parameters vary in
the topological phase diagram of the Floquet TFIM with homogeneous couplings.
These results are highlighted through examples, all chosen for numerical
convenience to be in one spatial dimension: non-integrable Floquet spin $1/2$
chains and Floquet $Z_3$ clock model where the latter hosts period-tripled edge
modes.
