ribbon category

**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
## 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

A **ribbon category** (also called a **tortile category**) is a braided pivotal category, or equivalently a balanced autonomous category, which satisfies $\theta^* = \theta$, where $\theta$ is the twist. This is a kind of category with duals.

Revised on June 2, 2010 21:35:21
by Mike Shulman
(128.135.72.152)