146
6 Operads
defined for m, n ≥ 0, 1 ≤ i ≤ m + 1, 1 ≤ j ≤ n + 1, g 1 , g 2 ∈ A, and degree 0
morphisms (the contractions)
◦ ij = ◦ ji : M (n + 2; g + s) → M (n; g)
defined for n ≥ 0, 1 ≤ i = j ≤ n and g ∈ A. These data are required to satisfy
the obvious modular versions of axioms (i)–(iii) of Definition 6.4 involving the i ◦ j -
operations, plus the following ones.
(i) For each n ≥ 0, g ∈ A, x ∈ M (n + 2; g) and a permutation ρ ∈ Σ n+2 ,
◦ ρ(i)ρ(j) (ρx) = λ◦ ij (x),
where λ ∈ Σ n is the composition
[n]
τ
−1
ij
−→ [n + 2] \ {i, j }
ρ
−→ [n + 2] \ {ρ(i), ρ(j )}
τ ρ(i)ρ(j)
−−−→ [n].
(ii) For m ≥ 0, g ∈ A, x ∈ M (m + 4; g), 1 ≤ c < d ≤ m + 4 and 1 ≤ a < b ≤
m + 2,
◦ ab ◦ cd (x) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
◦ c−2,d−2 ◦ ab (x),
if b < c,
◦ c−1,d−2 ◦ a,b+1 (x), if a < c ≤ b < d − 1,
◦ c−1,d−1 ◦ a,b+2 (x), if a < c, d − 1 ≤ b,
◦ c,d−2 ◦ a+1,b+1 (x), if c ≤ a < b < d − 1,
◦ c,d−1 ◦ a+1,b+2 (x), if c ≤ a < d − 1 ≤ b, and
◦ c,d ◦ a+2,b+2 (x),
if d − 1 ≤ a.
(iii) For m, n ≥ 0, g 1 , g 2 ∈ A, x ∈ M (m+2; g 1 ), y ∈ M (n+2; g 2 ), 1 ≤ a ≤ c−1
and c ≤ b ≤ c + n,
◦ ab (x c ◦ d y)
=
◦ c+n,a−b+c+n (x a ◦ b+d−c+1 y),
if c ≤ b < c − d + n + 2, and
◦ c+n,a−b+c+n (x a ◦ b+d−c−n−1 y), if c − d + n + 2 ≤ b ≤ c + n.
If c + n + 1 ≤ a ≤ m + n + 2 and x, y, b are as above, then
◦ ab (x c ◦ d y)
=
◦ c,a−b+c (x a−n ◦ b+d−c+1 y),
if c ≤ b < c − d + n + 2, and
◦ c,a−b+c (x a−n ◦ b+d−c−n−1 y), if c − d + n + 2 ≤ b ≤ c + n.