  Pursuing a generalization of group symmetries of modular categories to
category symmetries in topological phases of matter, we study linear Hopf
monads. The main goal is a generalization of extension and gauging group
symmetries to category symmetries of modular categories, which include also
categorical Hopf algebras as special cases. As an application, we propose an
analogue of the classification of finite simple groups to modular categories,
where we define simple modular categories as the prime ones without any
nontrivial normal algebras.
