# nLab category with duals

Categories with duals

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Categories 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.