category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A category with duals is a category where objects and/or morphisms have duals. This exists in several flavours; this list is mostly taken from a recent categories
list post from Peter Selinger.
A left autonomous category is a monoidal category in which every object is dualisable on the left.
A right autonomous category is a monoidal category in which every object is dualisable on the right.
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 $A$ to $A^{**}$, 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.
A spherical category is a pivotal category where the left and right trace operations coincide on all objects.
A tortile category, or ribbon category, is a balanced autonomous (therefore pivotal) category in which the twist on $A^*$ is the dual of the twist on $A$.
A compact closed category is a symmetric tortile category, or equivalently, a symmetric autonomous category.
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.
categories
post of 2010-05-15;