Jump to content

Ribbon category

From Wikipedia, the free encyclopedia

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). That is, for any two objects , there is an object . 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 , and similarly with . The twists are maps


such that

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

Concrete Example


Consider the category of finite-dimensional vector spaces over . Suppose that is such a vector space, spanned by the basis vectors . We assign to the dual object spanned by the basis vectors . Then let us define

and its dual

(which largely amounts to assigning a given the dual ).

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 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. But any object can be written in the form for some , , 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 ). Let us for example take the braiding

Then . Since , then ; by induction, if is -dimensional, then .

Other Examples


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 †: CopC coherently preserves the ribbon structure.


  1. ^ Turaev 2020, XI. An algebraic construction of modular categories
  2. ^ Turaev 2020, p. 25
  • Turaev, V.G. (2020) [1994]. Quantum Invariants of Knots and 3-Manifolds. de Gruyter. ISBN 978-3-11-088327-5.
  • Yetter, David N. (2001). Functorial Knot Theory. World Scientific. ISBN 978-981-281-046-5. OCLC 1149402321.
  • Ribbon category at the nLab