Ribbon category

In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category.
Definition.
A monoidal category formula_1 is, loosely speaking, a category equipped with a notion resembling the tensor product (of vector spaces, say). That is, for any two objects formula_2, there is an object formula_3. The assignment formula_4 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 formula_6 there is another object (called the left dual), formula_7, with maps
such that the compositions
equals the identity of formula_7, and similarly with formula_6. 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 †: Cop → C coherently preserves the ribbon structure.