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Remarks 4.1.2.
1. Graphically, we represent the braiding by overcrossings and its inverse by undercrossings.
Overcrossings and undercrossings have to be distinguished.
2. The flip map
τ : V ⊗ W → W ⊗ V
v ⊗ w ↦→ w ⊗ v
defines a symmetric braiding on the monoidal category vect(K) of K-vector spaces. It also
induces a symmetric braiding on the category K[G]-mod of K-linear representations of a
group. More generally, flip maps give a symmetric braiding on the category H−mod for
any cocommutative Hopf algebra. Since the universal enveloping algebra U(g) of a Lie
algebra g is cocommutative, the category of K-linear representations of g is symmetric,
as well.
3. There are tensor categories that do not admit a braiding. For example, for G a non-abelian
group, the category vect(G) of G-graded vector spaces does not admit a braiding since
are not isomorphic, if gh ≠ hg.
K g ⊗ K h
∼ = Kgh and K h ⊗ K g
∼ = Khg
4. The category vect(G) admits the flip as a braiding, if the group G is abelian. In the
case of G = Z 2 , objects are Z 2 -graded vector spaces V 0 ⊕ V 1 . We can introduce another
symmetric braiding c: on homogeneous vector spaces, it equals the flip, except for
c : V 1 ⊗ W 1 → W 1 ⊗ V 1
v 1 ⊗ w 1 ↦→ −w 1 ⊗ v 1
This category is called the category of super vector spaces. In particular, a tensor category
can admit inequivalent braidings.
5. The monoidal category of cobordisms has disjoint union as the tensor product. It admits
the morphism represented by the bordism of two cylinders exchanging the order of the
two tensorands M 1
∐
M2 → M 2
∐
M1 as a symmetric braiding.
Remarks 4.1.3.
1. As in any tensor category, we can consider algebras and coalgebras in a braided tensor
category (C, ⊗, c). Now, we have the notion of a commutative associative unital algebra
(A, μ, η): here the product is required to obey μ ◦ c A,A = μ. We also have the opposed
algebra with multiplication μ opp := μ◦c A,A . Similarly, we have the notion of the coopposed
coalgebra with coproduct Δ copp := c C,C ◦Δ, and the notion of a cocommative coassociative
counital coalgebra with coproduct obeying c C,C ◦ Δ = Δ.
2. Another construction that uses the braiding is the following: Suppose that we have two
associative unital algebras (A, μ, η) and (A ′ , μ ′ , η ′ ) in a braided tensor category C. Then
the tensor product A ⊗ A ′ can be endowed with the structure of an associative algebra
with product
A ⊗ A ′ ⊗ A ⊗ A ′ id A⊗c A,A ′ ⊗id A ′
−→
A ⊗ A ⊗ A ′ ⊗ A ′ μ⊗μ′
−→ A ⊗ A ′ .
A unit is then η ⊗ η ′ .
Dually, also the tensor product of two counital coassociative coalgebras can be endowed
with the structure of a coalgebra.
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