Tannaka duality and convolution for duoidal categories

TL;DR

This paper explores the relationship between bimonoid structures and monoidal structures in duoidal categories using a Tannaka adjunction approach, unifying classical convolution concepts and introducing new categorical structures.

Contribution

It introduces a novel approach using hom-enriched categories and internal homs to analyze bimonoids and monoidal structures in duoidal categories, extending classical convolution theory.

Findings

Hom-functors are monoidal, preserving monoids under convolution.

Unified classical convolution and Day convolution within duoidal categories.

Defined Hopf bimonoids and constructed new duoidal categories through warped structures.

Abstract

Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the Tannaka adjunction. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts ("actegories"). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors are monoidal, and so take monoids to monoids, unifies classical convolution in algebra and Day convolution for categories. Hopf bimonoids are defined leading to a lifting of closed structures. Warped monoidal structures permit the construction of new duoidal categories.