# Skew monoidales, skew warpings and quantum categories

## 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.

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