  Anomalies of global symmetries are important tools for understanding the
dynamics of quantum systems. We investigate anomalies of non-invertible
symmetries in 3+1d using 4+1d bulk topological quantum field theories given by
Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging
the 0-form symmetry gives the 4+1d "inflow" symmetry topological field theory
for the non-invertible symmetry. We find a two levels of anomalies: (1) the
bulk may fail to have an appropriate set of loop excitations which can condense
to trivialize the boundary dynamics, and (2) the "Frobenius-Schur indicator" of
the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d
fusion categories) may be incompatible with trivial boundary dynamics. As a
consequence we derive conditions for non-invertible symmetries in 3+1d to be
compatible with symmetric gapped phases, and invertible gapped phases. Along
the way, we see that the defects characterizing $\mathbb{Z}_{4}$ ordinary
symmetry host worldvolume theories with time-reversal symmetry $\mathsf{T}$
obeying the algebra $\mathsf{T}^{2}=C$ or $\mathsf{T}^{2}=(-1)^{F}C,$ with $C$
a unitary charge conjugation symmetry. We classify the anomalies of this
symmetry algebra in 2+1d and further use these ideas to construct 2+1d
topological orders with non-invertible time-reversal symmetry that permutes
anyons. As a concrete realization of our general discussion, we construct new
lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain
their dynamics.
