  In axion-Maxwell theory at the minimal axion-photon coupling, we find
non-invertible 0- and 1-form global symmetries arising from the naive shift and
center symmetries. Since the Gauss law is anomalous, there is no conserved,
gauge-invariant, and quantized electric charge. Rather, using half higher
gauging, we find a non-invertible Gauss law associated with a non-invertible
1-form global symmetry, which is related to the Page charge. These symmetries
act invertibly on the axion field and Wilson line, but non-invertibly on the
monopoles and axion strings, leading to selection rules related to the Witten
effect. We also derive various crossing relations between the defects. The
non-invertible 0- and 1-form global symmetries mix with other invertible
symmetries in a way reminiscent of a higher-group symmetry. Using this
non-invertible higher symmetry structure, we derive universal inequalities on
the energy scales where different infrared symmetries emerge in any
renormalization group flow to the axion-Maxwell theory. Finally, we discuss
implications for the Weak Gravity Conjecture and the Completeness Hypothesis in
quantum gravity.
