nLab category with duals

Idea

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.

Categories with duals for objects

• 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.

Categories with duals for morphisms

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.

There at least two commonspread kinds of categories with duals for morphisms:

• dagger categories where each morphism $f:X \to Y$ has a $\dagger$-dual $f^\dagger : Y \to X$, without any extra property.
• groupoids, where each morphism $f:X \to Y$ has an inverse $f^{-1} :Y \to X$ defined by the properties $f f^{-1} = 1_Y$, $f^{-1}f = 1_X$.

Moreover, every category enriched in one of the kind of categories listed above will have a notion of ‘dual’ for its morphisms.

References

• Peter Selinger, categories post of 2010-05-15;
• Peter Selinger, A survey of graphical languages for monoidal categories