In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category.
lC
C1,C2\inlC
C1 ⊗ C2\inlC
C1,C2\mapstoC1 ⊗ C2
c | |
C1,C2 |
:C1 ⊗ C2\stackrel\cong → C2 ⊗ C1.
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
C*
1 → C ⊗ C*,C* ⊗ C → 1
C*\congC* ⊗ 1 → C* ⊗ (C ⊗ C*)\cong(C* ⊗ C) ⊗ C* → 1 ⊗ C*\congC*
C*
C
C\inlC
\thetaC:C → C
\begin{align}
\theta | |
C1 ⊗ C2 |
&=
c | |
C2,C1 |
c | |
C1,C2 |
(\theta | |
C1 |
⊗
\theta | |
C2 |
)\\ \theta1&=id\\
\theta | |
C* |
&=
*. \end{align} | |
(\theta | |
C) |
Consider the category
FdVect(C)
C
C
\hat{e1},\hat{e2}, … ,\hat{en}
C
C\dagger
\hat{e}1,\hat{e}2, … ,\hat{e}n
\begin{align} ⋅ : C\dagger ⊗ C&\to1\\
i ⋅ \hat{e | |
\hat{e} | |
j} |
&\mapsto\begin{cases}1&i=j\ 0&i ≠ j\end{cases} \end{align}
and its dual
\begin{align} kIn:1&\toC ⊗ C\dagger\\ k&\mapstok
n | |
\sum | |
i=1 |
\hat{ei} ⊗ \hat{e}i\\ &=\begin{pmatrix}k&0& … &0\ 0&k&&\vdots\ &&\ddots&\\0& … &&k\end{pmatrix} \end{align}
(which largely amounts to assigning a given
\hat{ei}
\hat{e}i
Then indeed we find that (for example)
\begin{align} \hat{e}i&\cong\hat{e}i ⊗ 1\\ &\underset{In}{\to}\hat{e}i ⊗
n | |
\sum | |
j=1 |
\hat{ej} ⊗ \hat{e}j\\ &\cong
n | |
\sum | |
j=1 |
\left(\hat{e}i ⊗ \hat{ej}\right) ⊗ \hat{e}j\\ &\underset{ ⋅ }{\to}
n | |
\sum | |
j=1 |
\begin{cases}1 ⊗ \hat{e}j&i=j\ 0 ⊗ \hat{e}j&i ≠ j\end{cases}\\ &=1 ⊗ \hat{e}i\cong\hat{e}i \end{align}
and similarly for
\hat{ei}
FdVect
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
\begin{align} c | |
C1,C2 |
:C1 ⊗ C2&\toC2 ⊗ C1\\
c | |
C1,C2 |
(a,b)&\mapsto(b,a) \end{align}
then
c | |
C1,C2 |
c | |
C2,C1 |
=id | |
C1 ⊗ C2 |
\theta | |
C1 ⊗ C2 |
=
\theta | |
C1 |
⊗
\theta | |
C2 |
C\inFdVect
n | |
C=otimes | |
i=1 |
1
n
\thetaC=otimes
n | |
i=1 |
\theta1=
n | |
otimes | |
i=1 |
id=idC
On the other hand, we can introduce any nonzero multiplicative factor into the above braiding rule without breaking isomorphism (at least in
C
\begin{align} c | |
C1,C2 |
:C1 ⊗ C2&\toC2 ⊗ C1\\
c | |
C1,C2 |
(a,b)&\mapstoi(b,a) \end{align}
Then
c | |
C1,C2 |
c | |
C2,C1 |
=-id | |
C1 ⊗ C2 |
\theta1=id
\theta1=-id1 ⊗
C
n
\thetaC=(-1)n+1idC
The name ribbon category is motivated by a graphical depiction of morphisms.
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.