# Tannaka duality and convolution for duoidal categories

## Thomas Booker and Ross Street

Given a horizontal monoid $M$ in a duoidal category $\cal F$, we examine the relationship between bimonoid structures on $M$ and monoidal structures on the category $\cal F^{\ast M}$ of right $M$-modules which lift the vertical monoidal structure of $\cal F$. 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 $\cal F$ 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 $\cal F$ to $\cal F^{\ast M}$. We introduce the concept of warping monoidal structures and this permits the construction of new duoidal categories.

Keywords: duoidal; duoid; bimonoid; Tannaka duality; monoidal category; closed category; Hopf monoid

2010 MSC: 18D35; 18D10; 20J06

Theory and Applications of Categories, Vol. 28, 2013, No. 6, pp 166-205.

Published 2013-04-01.

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