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Skew monoidales, skew warpings and quantum categories

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Stephen Lack and Ross Street

Kornel Szlachanyi recently used the term skew-monoidal category for a
particular laxified version of monoidal category. He showed that
bialgebroids $H$ with base ring $R$ could be characterized in terms of
skew-monoidal structures on the category of one-sided $R$-modules for
which the lax unit was $R$ itself. We define skew monoidales (or skew
pseudo-monoids) in any monoidal bicategory $\cal M$. These are
skew-monoidal categories when $\cal M$ is $Cat$. Our main results are
presented at the level of monoidal bicategories. However, a consequence is
that quantum categories with base comonoid $C$ in a suitably complete
braided monoidal category $\CV$ are precisely skew monoidales in $Comod
(\cal V)$ with unit coming from the counit of $C$. Quantum groupoids (in
the sense of Chikhladze et al rather than Day and Street) are those skew
monoidales with invertible associativity constraint. In fact, we provide
some very general results connecting opmonoidal monads and skew
monoidales. We use a lax version of the concept of warping defined by
Booker and Street to modify monoidal structures.

Keywords:
bialgebroid; fusion operator; quantum category; monoidal bicategory;
monoidale; skew-monoidal category; comonoid; Hopf monad

2010 MSC:
18D10; 18D05; 16T15; 17B37; 20G42; 81R50

*Theory and Applications of Categories,*
Vol. 26, 2012,
No. 15, pp 385-402.

Published 2012-09-04.

http://www.tac.mta.ca/tac/volumes/26/15/26-15.dvi

http://www.tac.mta.ca/tac/volumes/26/15/26-15.ps

http://www.tac.mta.ca/tac/volumes/26/15/26-15.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/15/26-15.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/15/26-15.ps

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