6.1 Cyclic Operads
103
where λ ∈ Σ m+n is the composition
[m + n]
κ
−1
ij
−→
[m + 1] \ {i}
[n + 1] \ {j }
ρ σ
−−−→
[m + 1] \ {ρ(i)}
[n + 1] \ {σ (j)}
κ ρ(i)σ (j)
−−−−→ [m + n]
that involves the maps (6.7).
(ii) For x ∈ P(m + 1), y ∈ P(n + 1), 1 ≤ i ≤ m + 1, 1 ≤ j ≤ n + 1,
x i ◦ j y = (−1)
|x||y| λ(y j ◦ i x),
where λ ∈ Σ n+m is the cyclic permutation that takes j − i + m + 1 to 1.
(iii) For x ∈ P(n 1 + 1), y ∈ P(n 2 + 1), x ∈ P(n 3 + 1), 1 ≤ i ≤ n 1 + 1,
1 ≤ k ≤ n 2 + 1, 1 ≤ l ≤ n 3 + 1, 1 ≤ j ≤ n 2 + n 3 ,
x i ◦ j (y k ◦ l z)
=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(x i ◦ j y) k+i−j −1 ◦ l z,
j < k,
(−1) |x||y| λ·y k ◦ l (x i ◦ l+j −k+1 z),
k ≤ j < k + n 3 − l + 1,
(−1) |x||y| λ·y k ◦ i−j +k+n 3 −1 (x i ◦ l+j −k−n 3 z), k + n 3 − l + 1 ≤ j < k + n 3 ,
(x i ◦ j −n 3 +1 y) k ◦ l z,
k + n 3 ≤ j ≤ n 2 + n 3 ,
where λ is the cyclic permutation of [n 1 + n 2 + n 3 − 1] taking j + n 1 − i + 1
to 1.
Remark 6.5 Let P =
P(n) | n ≥ 0
be the skeletal presentation of a cyclic
operad with the structure operations i ◦ j as in Definition 6.4 above. Denote by Q =
Q(n) | n ≥ 0
the collection with Q(n) := P(n + 1). Then
◦ i : Q(m) ⊗ Q(n) → Q(m + n − 1), m, n ≥ 0,
defined, for 1 ≤ i ≤ m, by
x ◦ i y := x i ◦ n+1 y,
(6.10)
where x ∈ Q(m) = P(m + 1) and y ∈ Q(n) = P(n + 1), are the standard
◦ i -operations classically used to define operads, see [7, Definition 1.1] or [12,
Definition II.1.16].
Our definition of cyclic operads is slightly more general that the original one [2]
in that we admit nontrivial P(S) in Definition 6.1 for S the empty or a one-element
set, resp. nontrivial P(n) with n ≤ 1 in Definition 6.4. On the other hand, there