## 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 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.

Original language | English |
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Pages (from-to) | 166-205 |

Number of pages | 40 |

Journal | Theory and Applications of Categories |

Volume | 28 |

Publication status | Published - 1 Apr 2013 |