Skew monoidales, skew warpings and quantum categories
By Stephen Lack, Ross Street. Theory and Applications of Categories, 2012
47 citations recorded by the source corpus.
Kornel Szlach\'anyi 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 $\mathscr M$. These are skew-monoidal categories when $\mathscr M$ is $\mathrm{Cat}$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories in the sense of Day-Street with base comonoid $C$ in a suitably complete braided monoidal category $\mathscr V$ are precisely skew monoidales in $\mathrm{Comod} (\mathscr V)$ with unit coming from the counit of $C$. Quantum groupoids 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 recently by Booker-Street to modify monoidal structures.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2ec517d665c25ac5aa2841b20f84746e1b23317e
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In Algebra and Number Theory
In Geometry and Topology
- Ross Street
- Stephen Lack
- Michael Batanin
- Mark Weber
- Aaron D. Lauda
- Bob Coecke
- C. Berger
- Joachim Kock
In Mathematical Physics
- Ross Street
- Stephen Lack
- Michael Batanin
- Mark Weber
- Aaron D. Lauda
- Bob Coecke
- C. Berger
- Joachim Kock
Citations in the corpus
- Skew-closed categories (2012, background): “Apart from providing examples of skew-closed categories, the next result makes the connection with [1] and [9].”
- Monads need not be endofunctors (2010, background): “ty laws are not isomorphisms have been considered by multiple authors. Skew-monoidal categories were introduced by Szlacha´nyi [33] and caught then the interest of by Lack, Street, Buckley and Garner [22, 23, 12]. The axioms of skew-monoidality were also part of the axioms of the categories of conte”