6.4 Modular Operads
135
(iv) For mutually disjoint sets S 1 , S 2 , S 3 , symbols a, b, c, d, and genera
g 1 , g 2 , g 3 ∈ A, one has the equality
a ◦ b (1 ⊗ c ◦ d ) = c ◦ d ( a ◦ b ⊗1)
(6.55)
of maps from M
S 1 {a}; g 1
⊗ M
S 2 {b, c}; g 2
⊗ M
S 3 {d}; g 3
to
the space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
(v) For a finite set S, symbols a, b, c, d and a genus g ∈ A one has the equality
◦ ab ◦ cd = ◦ cd ◦ ab
(6.56)
of maps M
S {a, b, c, d}; g
→ M (S; g + 2s).
(vi) For finite sets S 1 , S 2 , symbols a, b, c, d, and genera g 1 , g 2 ∈ A, one has the
equality
◦ ab c ◦ d = ◦ cd a ◦ b
(6.57)
of maps M
S 1 {a, c}; g 1
⊗ M
S 2 {b, d}; g 2
→ M (S 1 S 2 ; g 1 + g 2 + s).
(vii) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a ◦ b (◦ uv ⊗ 1) = ◦ uv a ◦ b
(6.58)
of maps M
S 1 {a, u, v}; g 1
⊗ M
S 2 g 2
→ M (S 1 S 2 ; g 1 + g 2 + s).
Notice that the u ◦ v -operations preserve the A-grading, while the contractions ◦ uv
raise it by the step s. The existing definitions of modular operads as given, e.g., in [3]
have always assumed that A := N and s = 1. There are several situations where this
assumption is too restrictive.
Consider, for instance, two modular operads M and M with A := N and
s = 1. There exists an obvious product formula for the u ◦ v - and ◦ uv -operations on
the modular module M := M ⊗ M with
M (S; g) :=
g +g =g
M
(S; g
) ⊗ M (S; g
)
using those of M resp. M , but the contractions thus defined clearly raise the genus
grading by 2. To have a monoidal structure on the category of modular operads we
must thus allow arbitrary steps. The products of operads M and M with the steps
s and s , respectively, are then a modular operad with the step s + s . We will see
later that assuming A = N is also too restrictive.
Example 6.23 In all interesting examples of modular operad one has s = 0. It
is an easy exercise that then a modular operad for which M (S; g) = 0 only if
135
(iv) For mutually disjoint sets S 1 , S 2 , S 3 , symbols a, b, c, d, and genera
g 1 , g 2 , g 3 ∈ A, one has the equality
a ◦ b (1 ⊗ c ◦ d ) = c ◦ d ( a ◦ b ⊗1)
(6.55)
of maps from M
S 1 {a}; g 1
⊗ M
S 2 {b, c}; g 2
⊗ M
S 3 {d}; g 3
to
the space M
S 1 S 2 S 3 ; g 1 +g 2 +g 3
.
(v) For a finite set S, symbols a, b, c, d and a genus g ∈ A one has the equality
◦ ab ◦ cd = ◦ cd ◦ ab
(6.56)
of maps M
S {a, b, c, d}; g
→ M (S; g + 2s).
(vi) For finite sets S 1 , S 2 , symbols a, b, c, d, and genera g 1 , g 2 ∈ A, one has the
equality
◦ ab c ◦ d = ◦ cd a ◦ b
(6.57)
of maps M
S 1 {a, c}; g 1
⊗ M
S 2 {b, d}; g 2
→ M (S 1 S 2 ; g 1 + g 2 + s).
(vii) For finite sets S 1 , S 2 , symbols a, b, u, v, and genera g 1 , g 2 ∈ A, one has the
equality
a ◦ b (◦ uv ⊗ 1) = ◦ uv a ◦ b
(6.58)
of maps M
S 1 {a, u, v}; g 1
⊗ M
S 2 g 2
→ M (S 1 S 2 ; g 1 + g 2 + s).
Notice that the u ◦ v -operations preserve the A-grading, while the contractions ◦ uv
raise it by the step s. The existing definitions of modular operads as given, e.g., in [3]
have always assumed that A := N and s = 1. There are several situations where this
assumption is too restrictive.
Consider, for instance, two modular operads M and M with A := N and
s = 1. There exists an obvious product formula for the u ◦ v - and ◦ uv -operations on
the modular module M := M ⊗ M with
M (S; g) :=
g +g =g
M
(S; g
) ⊗ M (S; g
)
using those of M resp. M , but the contractions thus defined clearly raise the genus
grading by 2. To have a monoidal structure on the category of modular operads we
must thus allow arbitrary steps. The products of operads M and M with the steps
s and s , respectively, are then a modular operad with the step s + s . We will see
later that assuming A = N is also too restrictive.
Example 6.23 In all interesting examples of modular operad one has s = 0. It
is an easy exercise that then a modular operad for which M (S; g) = 0 only if