Skew monoidales, skew warpings and quantum categories

Stephen Lack*, Ross Street

*Corresponding author for this work

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    Kornel Szlachanyi [28] 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 [10] 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 [6] rather than [10]) 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 [3] to modify monoidal structures.

    Original languageEnglish
    Pages (from-to)385-402
    Number of pages18
    JournalTheory and Applications of Categories
    Issue number15
    Publication statusPublished - 2012

    Bibliographical note

    Copyright 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.


    • bialgebroid
    • fusion operator
    • quantum category
    • monoidal bicategory
    • monoidale
    • skew-monoidal category
    • comonoid
    • Hopf monad

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