## Extraordinary stuff for the antipodes

### Ross Street – 24 January 2001

This is joint work with Brian Day and Paddy McCrudden.
The definition and calculus of extraordinary natural transformations (in the sense of Eilenberg-Kelly) is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules. Since an exact pairing between an object and its dual is extraordinarily natural in the object, we are able to define dualization in any pseudomonoid in an autonomous monoidal bicategory directly generalizing the definition of an autonomous (= rigid) monoidal category. The antipode of a quasi-Hopf algebra H in the sense of Drinfeld is another example obtained by changing the monoidal bicategory; then, the reason the category Comodf(H) of finite dimensional representations of H is autonomous is that the Comodf operation preserves dualization.

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