(3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure ${\mathbb{Z}}_2$ gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure “spin-${\mathbb{Z}}_2$” gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion $2$-category ${\mathcal{Z}}_{(1)}(\mathrm{\Sigma }\mathbf{S}\mathbf{V}\mathbf{e}\mathbf{c})$ of string and particle operators in pure spin-${\mathbb{Z}}_2$ gauge theory.