6.1 Cyclic Operads
99
for x ∈ P
S 1 {a}
, y ∈ P
S 2 {b, c}
and z ∈ P
S 3 {d}
, can be written as
the associativity
x a ◦ b (y c ◦ d z) = (x a ◦ b y) c ◦ d z
(6.5)
of the contraction. Geometrically it means that, e.g., in Fig. 6.1, the result of the
contraction would not depend on whether we contract the edge e first and then f , or
vice versa.
Remark 6.2 We assume that S → P(S), ρ → P(ρ) is a covariant functor, so we
have the left actions in (6.9) of the skeletal version below. The conventions when
the assignment S → P(S) is contravariant and, therefore, (6.9) the right actions
are also sometimes used in the literature. The translation between these conventions
is straightforward though very technical.
Remark 6.3 The assumption that the sets S 1 and S 2 in (6.2) are disjoint is too
restrictive for some applications. The remedy is to use coproducts of sets instead
of their disjoint unions.
Recall that a coproduct of A 1 and A 2 is a set A 1 A 2 equipped with two
injections (coprojections) ι i : A i → A 1 A 2 , i = 1, 2. It is characterized by
the property that, for arbitrary set S, the assignment f → (f ι 1 , f ι 2 ) is a oneto-one correspondence between maps A 1 A 2 → S and couples (f 1 , f 2 ) of maps
f i : A i → S. The coproduct, defined up to a canonical isomorphism, can be realized
as the (ordinary) union of disjoint copies of A 1 and A 2 .
With these preliminaries, we define the extended a ◦ b -operations
a ◦ b : P
S 1 {a}
⊗ P
S 2 {b}
→ P(S 1 S 2 )
(6.6)
as the composition
P
S 1 {a}
⊗P
S 2 {b}
∼ =
−→ P
ι 1 (S 1 {a})
⊗P
ι 2 (S 2 {b})
ι 2 (a) ◦ ι 2 (b)
−−−−−−→ P
ι 1 (S 1 ) ι 2 (S 2 )
∼ =
−→ P(S 1 S 2 ),
where ι 1 , ι 2 are the coprojections for the coproduct
S 1 {a}
S 2 {b}
, the first
isomorphism is induced by the isomorphisms
S 1 {a}
∼ =
−→ ι 1
S 1 {(a)}
and S 2 {a}
∼ =
−→ ι 2
S 2 {(b)}
,
using the action (6.1), and the last isomorphism by the isomorphism
ι 1 (S 1 ) ι 2 (S 2 )
∼ =
−→ S 1 S 2
99
for x ∈ P
S 1 {a}
, y ∈ P
S 2 {b, c}
and z ∈ P
S 3 {d}
, can be written as
the associativity
x a ◦ b (y c ◦ d z) = (x a ◦ b y) c ◦ d z
(6.5)
of the contraction. Geometrically it means that, e.g., in Fig. 6.1, the result of the
contraction would not depend on whether we contract the edge e first and then f , or
vice versa.
Remark 6.2 We assume that S → P(S), ρ → P(ρ) is a covariant functor, so we
have the left actions in (6.9) of the skeletal version below. The conventions when
the assignment S → P(S) is contravariant and, therefore, (6.9) the right actions
are also sometimes used in the literature. The translation between these conventions
is straightforward though very technical.
Remark 6.3 The assumption that the sets S 1 and S 2 in (6.2) are disjoint is too
restrictive for some applications. The remedy is to use coproducts of sets instead
of their disjoint unions.
Recall that a coproduct of A 1 and A 2 is a set A 1 A 2 equipped with two
injections (coprojections) ι i : A i → A 1 A 2 , i = 1, 2. It is characterized by
the property that, for arbitrary set S, the assignment f → (f ι 1 , f ι 2 ) is a oneto-one correspondence between maps A 1 A 2 → S and couples (f 1 , f 2 ) of maps
f i : A i → S. The coproduct, defined up to a canonical isomorphism, can be realized
as the (ordinary) union of disjoint copies of A 1 and A 2 .
With these preliminaries, we define the extended a ◦ b -operations
a ◦ b : P
S 1 {a}
⊗ P
S 2 {b}
→ P(S 1 S 2 )
(6.6)
as the composition
P
S 1 {a}
⊗P
S 2 {b}
∼ =
−→ P
ι 1 (S 1 {a})
⊗P
ι 2 (S 2 {b})
ι 2 (a) ◦ ι 2 (b)
−−−−−−→ P
ι 1 (S 1 ) ι 2 (S 2 )
∼ =
−→ P(S 1 S 2 ),
where ι 1 , ι 2 are the coprojections for the coproduct
S 1 {a}
S 2 {b}
, the first
isomorphism is induced by the isomorphisms
S 1 {a}
∼ =
−→ ι 1
S 1 {(a)}
and S 2 {a}
∼ =
−→ ι 2
S 2 {(b)}
,
using the action (6.1), and the last isomorphism by the isomorphism
ι 1 (S 1 ) ι 2 (S 2 )
∼ =
−→ S 1 S 2