  We study the nonlinear $\sigma$-model in ${(d+1)}$-dimensional spacetime with
connected target space $K$ and show that, at energy scales below singularities
(e.g., vortices, hedgehogs, etc.), it has an emergent non-invertible higher
symmetry. The symmetry defects of the emergent symmetry are described by the
$d$-representations of a discrete $d$-group $\mathbb{G}^{(d)}$, so the emergent
symmetry is the dual of the invertible $d$-group $\mathbb{G}^{(d)}$ symmetry.
The $d$-group $\mathbb{G}^{(d)}$ is determined such that its classifying space
$B\mathbb{G}^{(d)}$ is given by the $d$-th Postnikov stage of $K$. In $(2+1)$D
for finite $\mathbb{G}^{(2)}$, this symmetry is always holo-equivalent to an
invertible ${0}$-form $\unicode{x2013}$ ordinary $\unicode{x2013}$ symmetry
with potential 't Hooft anomaly. The singularity-free disordered phase of the
nonlinear $\sigma$-model spontaneously breaks this symmetry, and when
$\mathbb{G}^{(d)}$ is finite, it is described by the deconfined phase of
$\mathbb{G}^{(d)}$ higher gauge theory. We consider examples of such disordered
phases. We focus on a singularity-free $S^2$ nonlinear $\sigma$-model in
${(3+1)}$D and show that its disordered phase is described by axion
electrodynamics and has two gapless modes corresponding to a photon and a
massless axion. This result is obtained via its emergent symmetry, which is the
dual of a $\mathbb{G}^{(3)}$ symmetry. Its classifying space
$B\mathbb{G}^{(3)}$ satisfies $\pi_1(B\mathbb{G}^{(3)})=0$,
$\pi_2(B\mathbb{G}^{(3)})=\mathbb{Z}$, $\pi_3(B\mathbb{G}^{(3)})=\mathbb{Z}$,
and has a nontrivial Postnikov invariant that describes the interplay between
$\pi_2$ and $\pi_3$. This means that the emergent symmetry is an exotic
continuous non-invertible symmetry in ${(3+1)}$D, while just an ordinary $U(1)$
invertible ${0}$-form symmetry in ${(2+1)}$D.
