On Étale Algebras and Bosonic Fusion 2-Categories

We classify all connected and Lagrangian étale algebras in the Drinfeld center ${\mathcal{Z}}_1({\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_G^{\pi })$, where $G$ is a finite group and $\pi$ is a 4-cocycle on $G$. By Décoppet’s result every bosonic fusion 2-category $\mathfrak{C}$ has its Drinfeld center equivalent to ${\mathcal{Z}}_1({\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_G^{\pi })$ for some $G$ and $\pi$. Combining this fact with classification of Lagrangian algebras in ${\mathcal{Z}}_1({\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_G^{\pi })$, we obtain a classification of bosonic fusion 2-categories.