Tannaka duality and convolution for duoidal categories
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.
Cite
@article{arxiv.1111.5659,
title = {Tannaka duality and convolution for duoidal categories},
author = {Thomas Booker and Ross Street},
journal= {arXiv preprint arXiv:1111.5659},
year = {2011}
}