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It defines a tensor product: given an associator a for ⊗, one verifies that a opp
U,V,W
:= a−1
W,V,U
is an
associator for the tensor product ⊗ opp . Similarly, one obtains left and right unit constraints.
Definition 4.1.1
1. A commutativity constraint for a tensor category (C, ⊗) is a natural isomorphism
c : ⊗ → ⊗ opp
of functors C × C → C. Explicitly, we have for any pair (V, W ) of objects of C an isomorphism
c V,W : V ⊗ W → W ⊗ V
such that for all morphisms V
commute.
f
−→ V ′ and W
V ⊗ W cV,W W ⊗ V
f⊗g
g
−→ W ′ the diagrams
g⊗f
V ′ ⊗ W ′ c V ′ ,W ′ W ′ ⊗ V ′
2. Let C be, for simplicity, a strict tensor category. A braiding is a commutatitivity constraint
such that for all objects U, V, W the compatibility relations with the tensor product
hold.
c U⊗V,W = (c U,W ⊗ id V ) ◦ (id U ⊗ c V,W )
c U,V ⊗W = (id V ⊗ c U,W ) ◦ (c U,V ⊗ id W )
If the category is not strict, the following two hexagon axioms have to hold:
c U,V ⊗W
U ⊗ (V ⊗ W )
a U,V,W
(U ⊗ V ) ⊗ W
c
U,V ⊗id W
(V ⊗ U) ⊗ W a V,U,W
(V ⊗ W ) ⊗
U
a V,W,U
V ⊗ (W ⊗ U)
id V ⊗c U,W V ⊗ (U ⊗ W )
and
(U ⊗ V ) ⊗ W cU⊗V,W W ⊗ (U ⊗ V )
a −1
a −1
U,V,W
W,U,V
U ⊗ (V ⊗ W )
(W ⊗ U) ⊗ V
id
U ⊗c V,W
a
−1
c U,W ⊗id V
U,W,V
U ⊗ (W ⊗ V )
(U ⊗ W ) ⊗ V
3. A braided tensor category is a tensor category together with the structure of a braiding.
4. With c UV , also c −1
V U is a braiding. In case, the identity c U,V = c −1
V,U
tensor category is called symmetric.
83
holds, the braided