Tannaka duality and convolution for duoidal categories

Thomas Booker ; Ross Street

Theory and applications of categories, Tome 28 (2013), pp. 166-205

Résumé

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.

Détail

Publié le : 2013-04-01

Classification : 18D35, 18D10, 20J06
Keywords: duoidal, duoid, bimonoid, Tannaka duality, monoidal category, closed category, Hopf monoid