Ribbon category

Definition

A monoidal category C is, loosely speaking, a category equipped with a notion resembling the tensor product (of vector spaces, say). That is, for any two objects C 1 , C 2 ∈ C , there is an object C 1 ⊗ C 2 ∈ C . The assignment C 1 , C 2 ↦ C 1 ⊗ C 2 is supposed to be functorial and needs to require a number of further properties such as a unit object 1 and an associativity isomorphism. Such a category is called braided if there are isomorphisms

A braided monoidal category is called a ribbon category if the category is left rigid and has a family of twists. The former means that for each object C there is another object (called the left dual), C ∗ , with maps

such that the compositions

equals the identity of C ∗ , and similarly with C . The twists are maps

such that

To be a ribbon category, the duals have to be compatible with the braiding and the twists in a certain way.

An example is the category of projective modules over a commutative ring. In this category, the monoidal structure is the tensor product, the dual object is the dual in the sense of (linear) algebra, which is again projective. The twists in this case are the identity maps. A more sophisticated example of a ribbon category are finite-dimensional representations of a quantum group.

The name ribbon category is motivated by a graphical depiction of morphisms.

Variant

A strongly ribbon category is a ribbon category C equipped with a dagger structure such that the functor †: C^op → C coherently preserves the ribbon structure.