# nLab category with duals

**monoidal categories** ## With symmetry * braided monoidal category * balanced monoidal category * twist * symmetric monoidal category ## With duals for objects * category with duals (list of them) * dualizable object (what they have) * rigid monoidal category, a.k.a. autonomous category * pivotal category * spherical category * ribbon category, a.k.a. tortile category * compact closed category ## With duals for morphisms * monoidal dagger-category? * symmetric monoidal dagger-category * dagger compact category ## With traces * trace * traced monoidal category ## Closed structure * closed monoidal category * cartesian closed category * closed category * star-autonomous category ## Special sorts of products * cartesian monoidal category * semicartesian monoidal category * multicategory ## Semisimplicity * semisimple category * fusion category * modular tensor category ## Morphisms * monoidal functor (lax, oplax, strong bilax, Frobenius) * braided monoidal functor * symmetric monoidal functor ## Internal monoids * monoid in a monoidal category * commutative monoid in a symmetric monoidal category * module over a monoid ## Examples * tensor product * closed monoidal structure on presheaves * Day convolution ## Theorems * coherence theorem for monoidal categories * monoidal Dold-Kan correspondence ## In higher category theory * monoidal 2-category * braided monoidal 2-category * monoidal bicategory * cartesian bicategory * k-tuply monoidal n-category * little cubes operad * monoidal (∞,1)-category * symmetric monoidal (∞,1)-category * compact double category

# 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.

## References

Revised on August 21, 2012 10:56:53 by Toby Bartels (98.19.40.130)