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 M. These are skew-monoidal categories when 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 V are precisely skew monoidales in Comod(V) with unit coming from the counit of C. Quantum groupoids (in the sense of  rather than ) 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 in  to modify monoidal structures.
|Number of pages||18|
|Journal||Theory and Applications of Categories|
|Publication status||Published - 2012|
Bibliographical noteCopyright the Author(s) 2012. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.
- fusion operator
- quantum category
- monoidal bicategory
- skew-monoidal category
- Hopf monad
- HOPF ALGEBROIDS