122
6 Operads
To define the a ◦ b -operations, we recall the gluing of Definition 6.9 and notice
that, by Lemma 6.2, one has for arbitrary trees T 1 and T 2 , the canonical isomorphisms
E(T 1 a ◦ b T 2 ) ∼ = E(T 1 ) ⊗ (T 2 )
which induce morphisms (in fact, isomorphisms) of the quotients
a ◦ b : F(E)
S 1 {a}
⊗ F(E)
S 2 {b}
→ F(E)
S 1 S 2
.
It is simple to prove that the above operations make F(E) a cyclic operad. It is stable
if and only if E(S) = ∅ for each S with less than three elements.
Proposition 6.2 The cyclic operad F(E) is free on the cyclic module E, i.e., for any
cyclic operad P and any morphism of cyclic modules f : E → P 4 there exists
a unique operad morphism Φ : F(E) → P such that f = Φ ◦ ι, where ι is the
obvious inclusion E E→ F(E) of cyclic modules. In diagrams:
E
(E)
ι
f
Φ
Proof. The proof follows a standard scheme, cf. [12, Proposition II.1.92], which we
will not reproduce here.
The operad morphism Φ : F(E) → P of Proposition 6.2 is usually called the
extension of f : E → P. A concise categorical reformulation of Proposition 6.2 is
that the functor
F : CycMod −→ CycOp, E −→ F(E)
is a left adjoint of the forgetful functor (6.31) which, by definition [6, p. 38], means
the existence of a natural isomorphism of the morphism spaces
CycMod
E,
∼ = CycOp
F(E), P
.
In the following text we will again make no notational distinction between
a cyclic operad P and its underlying cyclic module; it will always be clear what
we mean. Let
Π : F(P) → P
(6.36)
4 More precisely, f : E → but the implicit presence of the box is clear from the context.