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6 Operads
Remark 6.10 Later on, we will also need degree-k morphisms of modular modules
for an integer k. It is a family as in (6.49), but this time consisting of morphisms of
degree k. For k = 0 this extended notion of course agrees with Definition 6.15.
Definition 6.16 Let s ∈ A be a chosen element called the step. A modular operad
with step s is a modular module
M =
M (S; g) ∈ Chain | (S; g) ∈ Cor × A
(6.50)
together with degree 0 morphisms (compositions)
a ◦ b : M
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M (S 1 S 2 ; g 1 + g 2 )
(6.51)
defined for arbitrary disjoint finite sets S 1 , S 2 , symbols a, b, and arbitrary genera
g 1 , g 2 ∈ A. There are, moreover, degree 0 contractions
◦ uv = ◦ vu : M
S {u, v}; g
→ M (S; g + s)
(6.52)
given for any finite set S, genus g ∈ A, and symbols u, v. These data are required to
satisfy the following axioms.
(i) For arbitrary isomorphisms ρ : S 1 {a} → T 1 and σ : S 2 {b} → T 2 of finite
sets and genera g 1 , g 2 ∈ A, one has the equality
M
ρ| S 1 σ | S 2
a ◦ b = ρ(a) ◦ σ (b)
M (ρ) ⊗ M (σ )
of maps
M
S 1 {a}; g 1
⊗ M
S 2 {b}; g 2
→ M
T 1 T 2 \ {ρ(a), σ (b)}; g 1 + g 2
.
(ii) For an isomorphism ρ : S {u, v} → T of finite sets and a genus g ∈ A, one
has the equality
M
ρ| S
◦ uv = ◦ ρ(u)ρ(v) M (ρ)
(6.53)
of maps M
S {u, v}; g
→ M
T \ {ρ(u), ρ(v)}; g + s
.
(iii) For S 1 , S 2 , a, b and g 1 , g 2 as in (6.51), one has the equality
a ◦ b = b ◦ a τ
(6.54)
of maps M (S 1 {a}; g 1 ) ⊗ M (S 2 {b}; g 2 ) → M
S 1 S 2 ; g 1 + g 2
. 9
9 Recall that τ is the commutativity constraint in the category of graded vector spaces.