Filtered Frobenius Algebras in Monoidal Categories

Abstract

Abstract We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale’s 1967 result: a filtered deformation of a Frobenius algebra over a field is Frobenius as well. Toward the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni and Menini, Galatius et al., and Gwillian and Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs and Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category $\mathcal {C}$ is represented by a Frobenius algebra in $\mathcal {C}$. Several directions for further investigation are also proposed.