Frobenius Monoidal Functors of Dijkgraaf–Witten Categories and Rigid Frobenius Algebras

We construct a separable Frobenius monoidal functor from \smash{$\mathcal{Z}\big(\mathsf{Vect}_H^{\omega|_H}\big)$} to $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$, recovering the classification of étale algebras in these categories by Davydov–Simmons [J. Algebra 471 (2017), 149–175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov–Ostrik [Adv. Math. 171 (2002), 183–227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz–Walton [Comm. Math. Phys., {t}o appear, arXiv:2202.08644] in the general case.