Non-Invertible Symmetries and Phase Transitions in Three Dimensions*

Quantum systems in 3+1-dimension that are invariant under gauging a $G^{(1)}$ one-form symmetry contain non-invertible topological defects which can be obtained by gauging in half of spacetime. We show that a symmetry preserving gapped phase is incongruous with the existence of certain duality defects. These duality defects are realized in Abelian lattice gauge theories and hence elucidate the nature of phase transitions in these systems.