In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category.
A monoidal category
is, loosely speaking, a category equipped with a notion resembling the tensor product (of vector spaces, say).
The assignment
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
there is another object (called the left dual),
, with maps such that the compositions equals the identity of
The twists are maps such that To be a ribbon category, the duals have to be thus compatible with the braiding and the twists.
{\displaystyle \mathbf {FdVect} (\mathbb {C} )}
of finite-dimensional vector spaces over
is such a vector space, spanned by the basis vectors
the dual object
spanned by the basis vectors
Then let us define and its dual (which largely amounts to assigning a given
Then indeed we find that (for example) and similarly for
Since this proof applies to any finite-dimensional vector space, we have shown that our structure over
{\displaystyle \mathbf {FdVect} }
defines a (left) rigid monoidal category.
Then, we must define braids and twists in such a way that they are compatible.
In this case, this largely makes one determined given the other on the reals.
For example, if we take the trivial braiding then
, so our twist must obey
In other words it must operate elementwise across tensor products.
can be written in the form
, so our twists must also be trivial.
On the other hand, we can introduce any nonzero multiplicative factor into the above braiding rule without breaking isomorphism (at least in
The name ribbon category is motivated by a graphical depiction of morphisms.
[2] 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.