Nondegenerate module categories

Abstract: Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is "nondegenerate" and "factorizable", and establish that these properties are equivalent. The proof involves a new monadicity result for module categories. Lastly, we examine the Hopf case, using Kolb's (2020) notion of a quasitriangular comodule algebra to introduce "factorizable" comodule algebras. We then show that the representation category of a quasitriangular comodule algebra is nondegenerate in our sense precisely when the comodule algebra is factorizable. Several examples are provided.