An autonomous category, or rigid category is a monoidal category that is both left and right autonomous. Note that any braided monoidal category is autonomous on both sides if it is autonomous on either side.
A pivotal category is an autonomous category equipped with a monoidal natural isomorphism from the identity functor to the double dual? functor. A one-sided autonomous category with such an isomorphism is automatically two-sided autonomous. Although each braided autonomous category has an isomorphism from to , such a category is not necessarily pivotal because this isomorphism is not in general monoidal. On the other hand, every balanced autonomous category is pivotal.
The -autonomous categories do not really belong on this list; being -autonomous is logically independent of being autonomous, and while -autonomous categories have duals, these are not in general duals in the sense of a dualisable object. However, any compact closed category is -autonomous.
Likewise, closed categories or closed monoidal categories do not really belong on this list, but there is a sense of dual there which should be carefully distinguished from the primary sense here, which is generally stronger. See dual object in a closed category.
One might write something about these too, or put them on a separate page. In the meantime, see the table of contents to the right.
categoriespost of 2010-05-15;