136
6 Operads
card(S) = 2 and g = 0 is precisely a graded (non-unital) associative algebra A with
an involution τ : A → A such that τ (ab) = τ (b)τ (a) for all a, b ∈ A.
In the seminal paper [3] where modular operads were introduced, the following
property of modular operads was always assumed.
Definition 6.17 A modular operad with A = N and s = 1 is stable if
M (S; g) = 0 for card(S) ≤ 2, g = 0 and for card(S) = 0, g = 1.
(6.59)
Using the notation
S :=
(S, g) | g ≥ 2, or g = 1 and card(S) ≥ 1, or g = 0 and card(S) ≥ 3
,
(6.60)
the stability of a modular operad M can be expressed by witting
M = {M (S; g) ∈ Chain | (S, g) ∈ S}.
Example 6.24 As cyclic operads, also modular operads exist in an arbitrary symmetric monoidal category, e.g., in the cartesian monoidal category Set of sets. One
has the terminal stable Set-modular operad ∗ Mod defined by
∗ Mod (S; g) :=
∗ if (S, g) ∈ S, and
∅ otherwise,
where ∗ is a chosen one-point set. The terminal stable modular operad ∗ Mod has a
geometric interpretation extending the interpretation of the terminal cyclic operad
given in Example 6.3.
Namely, for (S, g) ∈ S consider the set M(S; g) of isomorphism classes of
oriented closed surfaces of genus g with holes indexed by S. It is a stable modular
operad with a ◦ b as in Example 6.3, while ◦ uv is given by attaching a handle as
follows.
Let P be a surface of genus g with holes labeled by S {u, v}. We let ◦ uv (P ) to
denote the surface obtained from P by adding a tube connecting the circumference
of the hole labeled u with the circumference of the hole labeled v. This gives rise to
an operation
◦ ab : M
S {u, v}; g
→ M(S; g + 1)
on the set of isomorphism classes. Since there is for (S, g) ∈ S only one
isomorphism class in M(S; g), one sees that M is isomorphic to the terminal stable
modular operad ∗ Mod .
6 Operads
card(S) = 2 and g = 0 is precisely a graded (non-unital) associative algebra A with
an involution τ : A → A such that τ (ab) = τ (b)τ (a) for all a, b ∈ A.
In the seminal paper [3] where modular operads were introduced, the following
property of modular operads was always assumed.
Definition 6.17 A modular operad with A = N and s = 1 is stable if
M (S; g) = 0 for card(S) ≤ 2, g = 0 and for card(S) = 0, g = 1.
(6.59)
Using the notation
S :=
(S, g) | g ≥ 2, or g = 1 and card(S) ≥ 1, or g = 0 and card(S) ≥ 3
,
(6.60)
the stability of a modular operad M can be expressed by witting
M = {M (S; g) ∈ Chain | (S, g) ∈ S}.
Example 6.24 As cyclic operads, also modular operads exist in an arbitrary symmetric monoidal category, e.g., in the cartesian monoidal category Set of sets. One
has the terminal stable Set-modular operad ∗ Mod defined by
∗ Mod (S; g) :=
∗ if (S, g) ∈ S, and
∅ otherwise,
where ∗ is a chosen one-point set. The terminal stable modular operad ∗ Mod has a
geometric interpretation extending the interpretation of the terminal cyclic operad
given in Example 6.3.
Namely, for (S, g) ∈ S consider the set M(S; g) of isomorphism classes of
oriented closed surfaces of genus g with holes indexed by S. It is a stable modular
operad with a ◦ b as in Example 6.3, while ◦ uv is given by attaching a handle as
follows.
Let P be a surface of genus g with holes labeled by S {u, v}. We let ◦ uv (P ) to
denote the surface obtained from P by adding a tube connecting the circumference
of the hole labeled u with the circumference of the hole labeled v. This gives rise to
an operation
◦ ab : M
S {u, v}; g
→ M(S; g + 1)
on the set of isomorphism classes. Since there is for (S, g) ∈ S only one
isomorphism class in M(S; g), one sees that M is isomorphic to the terminal stable
modular operad ∗ Mod .