  We explore generalized global symmetries in theories of physics beyond the
Standard Model. Theories of $Z'$ bosons generically contain 'non-invertible'
chiral symmetries, whose presence indicates a natural paradigm to break this
symmetry by an exponentially small amount in an ultraviolet completion. For
example, in models of gauged lepton family difference such as the
phenomenologically well-motivated $U(1)_{L_\mu - L_\tau}$, there is a
non-invertible lepton number symmetry which protects neutrino masses. We embed
these theories in gauged non-Abelian horizontal lepton symmetries, e.g.
$U(1)_{L_\mu - L_\tau} \subset SU(3)_H$, where the generalized symmetries are
broken nonperturbatively by the existence of lepton family magnetic monopoles.
In such theories, either Majorana or Dirac neutrino masses may be generated
through quantum gauge theory effects from the charged lepton Yukawas e.g.
$y_\nu \sim y_\tau \exp(-S_{\rm inst})$. These theories require no bevy of new
fields nor ad hoc additional global symmetries, but are instead simple,
natural, and predictive: the discovery of a lepton family $Z'$ at low energies
will reveal the scale at which $L_\mu - L_\tau$ emerges from a larger gauge
symmetry.
