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6 Operads
using (6.1) again. Notice that P(S 1 S 2 ) as well as the map a ◦ b in (6.6) are defined
only up to functorial canonical isomorphisms.
The composition operations in Definition 6.1 were maps
a ◦ b : P
S 1 {a}
⊗ P
S 2 {b}
→ P(S 1 S 2 ).
It is sometimes convenient to consider an equivalent family of compositions, namely
a ◦ b : P(D 1 ) ⊗ P(D 2 ) → P
D 1 D 2 \ {a, b}
with D 1 := S 1 {a} and D 2 := S 2 {b}. One can easily verify that Definition 6.1
is equivalent to
Definition 6.3 A cyclic operad P is a family
P =
P(S) | S ∈ Cor
of dg-vector spaces together with degree 0 morphisms
P(ρ) : P(S) → P(D)
given for any isomorphism ρ : S → D of finite sets, and degree 0 morphisms
(compositions)
a ◦ b : P(S 1 ) ⊗ P(S 2 ) → P
S 1 S 2 \ {a, b}
defined for arbitrary disjoint finite sets S 1 , S 2 with elements a ∈ S 1 , b ∈ S 2 . These
data are required to satisfy the following axioms.
(i) One has P(1 S ) = 1 P(S) for any finite set S, and P(ρσ ) = P(ρ)P(σ ) for
arbitrary composable morphisms ρ, σ in Cor.
(ii) For arbitrary isomorphisms ρ : S 1 → T 1 and σ : S 2 → T 2 of finite sets, one
has the equality
P
ρ| S 1 \{a} σ | S 2 \{b}
a ◦ b = ρ(a) ◦ σ (b)
P(ρ) ⊗ P(σ )
of maps P(S 1 ) ⊗ P(S 2 ) → P
T 1 T 2 \ {ρ(a), σ (b)}
.
(iii) Let τ : P(S 1 ) ⊗ P(S 2 ) → P(S 2 ) ⊗ P(S 1 ) be the commutativity constraint
in (3). One requires the equality
a ◦ b = b ◦ a τ
of maps P(S 1 ) ⊗ P(S 2 ) → P
S 2 S 1 \ {a, b}).