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Tannaka duality and convolution for duoidal categories

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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.

http://www.tac.mta.ca/tac/volumes/28/6/28-06.dvi

http://www.tac.mta.ca/tac/volumes/28/6/28-06.ps

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