104
6 Operads
is an important class of cyclic operads whose components vanish on sets of small
cardinalities:
Definition 6.5 A cyclic operad P as in Definition 6.1 is stable if P(S) = 0 for
all S ∈ Cor with card(S) ≤ 2. In the skeletal setup of Definition 6.4 the stability
means that P(n) = 0 for n ≤ 2.
The above terminology is motivated by the stability property of smooth complex
projective curves of genus zero with n marked points. Such a curve is, by definition,
stable, if it has no infinitesimal automorphism fixing the marked points. It is wellknown that this happens if and only if n ≥ 3. A generalization of this notion to
arbitrary genera is recalled in Example 6.25.
Example 6.1 The cyclic operad Com is defined by
Com(S) :=
k if S has at least 3 elements, and
0 if S has less than 3 elements.
The functorial isomorphisms Com(σ ) : Com(S) → Com(D) are the identities and
the compositions
a ◦ b : Com
S 1 {a}
⊗ Com
S 2 {b}
−→ Com
S 1 S 2
the canonical isomorphisms k ⊗ k ∼ = k, k ⊗ 0 ∼ = 0, 0 ⊗ k ∼ = 0 or 0 ⊗ 0 ∼ = 0,
depending on the cardinalities of the sets S 1 resp. S 2 .
For a set S with at least 3 elements denote by μ S ∈ Com(S) the element
corresponding to 1 ∈ k = Com(S); for S with less than 3 elements we put μ S := 0.
It is clear from definition that
Com(σ )(μ S ) = μ D
for any isomorphism σ : S → D of finite sets and that
μ S 1 a ◦ b μ S 2 = μ S 1 2
(6.11)
for arbitrary disjoint finite sets S 1 and S 2 . The operad Com is stable. An example of
a non-stable cyclic operad is provided by the endomorphism operad End V recalled
in Example 6.6 below.
It should be clear that the cyclic operad Com can equivalently be defined by
Com(S) := Span(μ S ) for arbitrary finite set S, with the structure operations given
by (6.11). The third kind of definition via the generating operation and a relation
will be given in Proposition 6.4.
6 Operads
is an important class of cyclic operads whose components vanish on sets of small
cardinalities:
Definition 6.5 A cyclic operad P as in Definition 6.1 is stable if P(S) = 0 for
all S ∈ Cor with card(S) ≤ 2. In the skeletal setup of Definition 6.4 the stability
means that P(n) = 0 for n ≤ 2.
The above terminology is motivated by the stability property of smooth complex
projective curves of genus zero with n marked points. Such a curve is, by definition,
stable, if it has no infinitesimal automorphism fixing the marked points. It is wellknown that this happens if and only if n ≥ 3. A generalization of this notion to
arbitrary genera is recalled in Example 6.25.
Example 6.1 The cyclic operad Com is defined by
Com(S) :=
k if S has at least 3 elements, and
0 if S has less than 3 elements.
The functorial isomorphisms Com(σ ) : Com(S) → Com(D) are the identities and
the compositions
a ◦ b : Com
S 1 {a}
⊗ Com
S 2 {b}
−→ Com
S 1 S 2
the canonical isomorphisms k ⊗ k ∼ = k, k ⊗ 0 ∼ = 0, 0 ⊗ k ∼ = 0 or 0 ⊗ 0 ∼ = 0,
depending on the cardinalities of the sets S 1 resp. S 2 .
For a set S with at least 3 elements denote by μ S ∈ Com(S) the element
corresponding to 1 ∈ k = Com(S); for S with less than 3 elements we put μ S := 0.
It is clear from definition that
Com(σ )(μ S ) = μ D
for any isomorphism σ : S → D of finite sets and that
μ S 1 a ◦ b μ S 2 = μ S 1 2
(6.11)
for arbitrary disjoint finite sets S 1 and S 2 . The operad Com is stable. An example of
a non-stable cyclic operad is provided by the endomorphism operad End V recalled
in Example 6.6 below.
It should be clear that the cyclic operad Com can equivalently be defined by
Com(S) := Span(μ S ) for arbitrary finite set S, with the structure operations given
by (6.11). The third kind of definition via the generating operation and a relation
will be given in Proposition 6.4.