Filtered Frobenius algebras in monoidal categories

TL;DR

This paper develops categorical techniques to show that filtered deformations of Frobenius algebras remain Frobenius, with applications to module categories over tensor categories.

Contribution

It introduces a monoidal associated graded functor and characterizes Frobenius algebras in monoidal categories, extending classical results to categorical settings.

Findings

Filtered deformations of Frobenius algebras are Frobenius.

Categorical Frobenius forms characterize Frobenius algebras.

Any exact module category over a symmetric finite tensor category is represented by a Frobenius algebra.

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. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-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-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.