Filtered Frobenius algebras in monoidal categories

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. We illustrate these results by discussing braided Clifford algebras, which are filtered deformations of Bespalov et al.'s braided exterior algebras, and show that these are Frobenius algebras in symmetric rigid monoidal categories. Several directions for further investigation are proposed.