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Proof.
Identifying I ∼ = I ⊗ I, we see
ϕ ◦ ϕ ′ = (ϕ ⊗ id I ) ◦ (id I ⊗ ϕ ′ ) = ϕ ⊗ ϕ ′
and
ϕ ′ ◦ ϕ = (id I ⊗ ϕ ′ ) ◦ (ϕ ⊗ id I ) = ϕ ⊗ ϕ ′ .
✷
Definition 5.1.7
Let C be a pivotal category.
1. Let X be an object of C and f ∈ End C (X). We define left and right traces:
Tr l : End C (X) → End C (I)
f ↦→ d V ◦ (id V ∗ ⊗ f) ◦ ˜b V
Tr r : End C (X) → End C (I)
f ↦→ ˜d V ◦ (f ⊗ id V ∗) ◦ b V
(Some authors call this the quantum trace or the categorical trace.)
2. One also defines left and right dimensions:
dim l X := Tr l id X and dim r X := Tr r id X .
Lemma 5.1.8.
Both traces have the following properties:
1. The traces are symmetric: for any pair of morphisms g : X → Y and f : Y → X in C, we
have
Tr l (gf) = Tr l (fg) and Tr r (gf) = Tr r (fg) .
2. We have
Tr l (f) = Tr r (f ∗ ) = Tr l (f ∗∗ )
for any endomorphism f, and similar relations with left and right trace interchanged.
3. Suppose that
α ⊗ id X = id X ⊗ α for all α ∈ End C (I) and all objects X ∈ C . (13)
Then the traces are multiplicative for the tensor product:
Tr l (f ⊗ g) = Tr l (f) ∙ Tr l (g) and Tr r (f ⊗ g) = Tr r (f) ∙ Tr r (g)
for all endomorphisms f, g.
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