6.1 Cyclic Operads
105
Remark 6.6 Definition 6.1 and the equivalent definitions that follow define cyclic
operads in the symmetric monoidal category Chain of dg-vector spaces and
their degree 0 morphisms. Cyclic operads can however be defined in an arbitrary
symmetric monoidal category, for instance, in the cartesian monoidal category Set
of sets. Such a Set-cyclic operad S is a collection
S =
S (S) ∈ Set | S ∈ Cor
of sets together with maps of sets
S (ρ) : S (S) → S (D)
as in (6.1) and compositions
a ◦ b : S
S 1 {a}
× S
S 2 {b}
→ S (S 1 S 2 )
satisfying the obvious analogs of the axioms of Definition 6.1. Stability of such
an operad means that S (S) = ∅ if card(S) ≤ 2. Each Set-cyclic operad S
determines a Chain-operad Span(S ) with
Span(S )(S) := Span
S (S)
, S ∈ Cor,
where Span
S (S)
is the linear span of the set S (S). We call Span(S ) the
linearization of the Set-operad S . It is clear that Span(S ) is stable if and only
if S is stable.
Example 6.2 The subcategory of stable cyclic Set-operads has the terminal object
∗ cyclic given by
∗ cyclic (S) =
∗ if card(S) ≥ 3 and
∅ otherwise,
where ∗ is a one-point set. All its structure operations a ◦ b : ∗ × ∗ → ∗ are the
isomorphisms ∗ × ∗ ∼ = ∗ and the action (6.1) is trivial. It is clear that for each stable
cyclic Set-operad S there exists a unique morphism S → ∗ cyclic , which means
that ∗ cyclic is indeed a terminal Set-operad. The cyclic operad Com of Example 6.1
is the linearization of this terminal operad, that is,
Com ∼ = Span(∗ cyclic ).
(6.12)
Example 6.3 There exists a simple geometric interpretation of the terminal stable
cyclic operad ∗ cyclic . For S ∈ Cor, card(S) ≥ 3, denote by M 0 (S) the set of
isomorphism classes of oriented closed surfaces of genus 0 with holes labeled by
S. An example of such a surface is given in Fig. 6.2-left. For card(S) ≤ 2 put
M 0 (S) := ∅.
105
Remark 6.6 Definition 6.1 and the equivalent definitions that follow define cyclic
operads in the symmetric monoidal category Chain of dg-vector spaces and
their degree 0 morphisms. Cyclic operads can however be defined in an arbitrary
symmetric monoidal category, for instance, in the cartesian monoidal category Set
of sets. Such a Set-cyclic operad S is a collection
S =
S (S) ∈ Set | S ∈ Cor
of sets together with maps of sets
S (ρ) : S (S) → S (D)
as in (6.1) and compositions
a ◦ b : S
S 1 {a}
× S
S 2 {b}
→ S (S 1 S 2 )
satisfying the obvious analogs of the axioms of Definition 6.1. Stability of such
an operad means that S (S) = ∅ if card(S) ≤ 2. Each Set-cyclic operad S
determines a Chain-operad Span(S ) with
Span(S )(S) := Span
S (S)
, S ∈ Cor,
where Span
S (S)
is the linear span of the set S (S). We call Span(S ) the
linearization of the Set-operad S . It is clear that Span(S ) is stable if and only
if S is stable.
Example 6.2 The subcategory of stable cyclic Set-operads has the terminal object
∗ cyclic given by
∗ cyclic (S) =
∗ if card(S) ≥ 3 and
∅ otherwise,
where ∗ is a one-point set. All its structure operations a ◦ b : ∗ × ∗ → ∗ are the
isomorphisms ∗ × ∗ ∼ = ∗ and the action (6.1) is trivial. It is clear that for each stable
cyclic Set-operad S there exists a unique morphism S → ∗ cyclic , which means
that ∗ cyclic is indeed a terminal Set-operad. The cyclic operad Com of Example 6.1
is the linearization of this terminal operad, that is,
Com ∼ = Span(∗ cyclic ).
(6.12)
Example 6.3 There exists a simple geometric interpretation of the terminal stable
cyclic operad ∗ cyclic . For S ∈ Cor, card(S) ≥ 3, denote by M 0 (S) the set of
isomorphism classes of oriented closed surfaces of genus 0 with holes labeled by
S. An example of such a surface is given in Fig. 6.2-left. For card(S) ≤ 2 put
M 0 (S) := ∅.