6.1 Cyclic Operads
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given for any isomorphism ρ : S → T of finite sets, together with degree 0
morphisms (compositions)
a ◦ b : P
S 1 {a}
⊗ P
S 2 {b}
→ P(S 1 S 2 )
(6.2)
defined for arbitrary disjoint finite sets S 1 , S 2 and symbols a, b. 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 morphisms ρ : S 1 {a} → T 1 and σ : S 2 {b} → T 2 in Cor, one
has the equality
P
ρ| S 1 σ | S 2
a ◦ b = ρ(a) ◦ σ (b)
P(ρ) ⊗ P(σ )
of maps P(S 1 {a}) ⊗ P(S 2 {b}) → P
T 1 T 2 \ {ρ(a), ρ(b)}
.
(iii) Let τ : P
S 1 {a}
⊗ P
S 2 {b}
→ P
S 2 {b}
⊗ P
S 1 {a}
be the
flip (3). One then has the equality
a ◦ b = b ◦ a τ
of maps P
S 1 {a}
⊗ P
S 2 {b}
→ P(S 2 S 1 ). 1
(iv) For disjoint sets S 1 , S 2 , S 3 and symbols a, b, c, d one has the equality
a ◦ b (1 ⊗ c ◦ d ) = c ◦ d ( a ◦ b ⊗1)
(6.3)
of maps P
S 1 {a}
⊗ P
S 2 {b, c}
⊗ P
S 3 {d}
→ P(S 1 S 2 S 3 ).
The ambient category in which cyclic operads of Definition 6.1 live is the
category Chain of dg-vector spaces. This means that each P(S) is a graded vector
space with a degree +1 differential d = d P and that all structure operations of P
commute with the differentials. In particular, (6.2) is a map of dg-vector spaces,
where the tensor product P
S 1 {a}
⊗ P
S 2 {b}
bears the differential induced
in the standard manner. In elements this means that for x ∈ P
S 1 {a}
and
y ∈ P
S 2 {b}
,
d(x a ◦ b y) = (dx) a ◦ b y + (−1)
|x| x a ◦ b (dy).
(6.4)
To emphasize that P carries a differential, we will call such a P sometimes
more specifically a cyclic dg-operad. If we want to distinguish the differential d P
from other differentials that may occur in the same context, we will call it the
internal differential.
1 Since S 2 S 1 = S 1 S 2 , P(S 2 S 1 ) = P(S 1 S 2 ).