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6 Operads
In the above display, ¯
s and ¯ ¯
s are two copies of the map s. It is simple to verify that
both expressions above in fact equal
1 S 1 ⊗ ¯
s ⊗ 1 S 2 ⊗ ¯ ¯
s ⊗ 1 S 3
# (f ⊗ g ⊗ h)
which establishes (6.5). We shall however keep in mind that calculations using the
shorthand (6.17) implicitly involve several canonical identifications and inclusions.
We leave as an exercise to verify that the collection End V = {End V (S) | S ∈
Cor} with the above operations fulfills also the remaining axioms of cyclic operads.
The symmetry of s is necessary for axiom (iii) to hold. It is a generic example of a
cyclic operad in that all the axioms can be read off from it.
Definition 6.7 The operad End V , or End (V ,s) if we want to stress the rôle of the
symmetric tensor s, is called the cyclic endomorphism operad of the vector space V .
Example 6.7 We are going to describe a dual version of the endomorphism operad
End V of Example 6.6. This time V is a graded vector space equipped with a degree
0 symmetric, not necessarily non-degenerate bilinear form B : V ⊗V → k. For a
finite set S define
Dne V (S) :=
c∈S V c ,
where V c := V for each c ∈ S. Given an isomorphism σ : S → D of finite sets, put
Dne V (σ ) := σ : Dne V (S) → Dne V (D),
with σ as in (6.14). Let S 1 , S 2 be disjoint finite sets and a = b two symbols. Define
a ◦ b : Dne V
S 1 {a}
⊗ Dne V
S 2 {b}
→ Dne V
S 1 S 2
as the composition
Dne V
S 1 {a}
⊗ Dne V
S 2 {b}
∼ = Dne V
S 1 ) ⊗ V ⊗ V ⊗ Dne V
S 2 )
1⊗B⊗1
−−−→ Dne V
S 1 ) ⊗ Dne V
S 2 ) ∼ = Dne V
S 1 S 2 )
in which the isomorphisms are those of Lemma 6.2. In the shorthand similar
to (6.17) we may write
x a ◦ b y := (1 S 1 ⊗ B ⊗ 1 S 2 )(x ⊗ y).
We leave again as an exercise to verify that the collection Dne V = {Dne V (S) | S ∈
Cor} with the above operations is a cyclic operad.