6.4 Modular Operads
145
Proposition 6.10 The modular operad F(E) is the free modular operad on the
modular module E.
We finish this section by translating Definition 6.16 of modular operads into the
skeletal language. Recall that [n] := {1, . . . , n}, with [0] the empty set ∅. For M as
in (6.50), n ≥ 0 and g ∈ A denote M (n; g) := M ([n]; g). The operations
i ◦ j : M (m + 1; g 1 ) ⊗ M (n + 1; g 2 ) → M (m + n; g 1 + g 2 )
are, for 1 ≤ i ≤ m + 1, 1 ≤ j ≤ n + 1 and g 1 , g 2 ∈ A, given by the obvious analog
of formula (6.8). To define the skeletal version
◦ ij : M (n + 2; g + s) → M (n; g), 1 ≤ i, j ≤ n + 2, g ∈ A,
(6.68)
of the contractions ◦ uv in (6.52), we need an auxiliary map
τ = τ ij : [n + 2] \ {i, j } → [n]
(6.69)
given by
τ ij (a) :=
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
a,
for 1 ≤ a < i,
a − 1, for i < a < j, and
a − 2, for j < a ≤ n + 2,
if i < j, while for i > j we set τ ij := τ ji . Then ◦ ij in (6.68) is the composition
M (n + 2; g + s)
◦ ij
−→ M
[n + 2] \ {i, j }; g
M (τ ij )
−−−→ M (n; g),
(6.70)
where ◦ ij is the contraction (6.52) with S = [n+2]. An involved but straightforward
calculation shows that the above structure has the properties listed in the following
Definition 6.23 A modular operad M is a family
M = {M (n; g) | n ≥ 0, g ∈ A}
of dg-vector spaces together with linear left actions
Σ n × M (n; g) → M (n; g), n ≥ 1, g ∈ A,
of the symmetric groups Σ n , degree 0 morphisms
i ◦ j : M (m + 1; g 1 ) ⊗ M (n + 1; g 2 ) → M (m + n; g 1 + g 2 ),