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 F*M of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the so-called Tannaka adjunction; that is, an adjunction inducing the equivalence which expresses Tannaka duality. 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 on F to F*M. We introduce the concept of warping monoidal structures and this permits the construction of new duoidal categories.
|Number of pages||40|
|Journal||Theory and Applications of Categories|
|Publication status||Published - 1 Apr 2013|