
156 Algebraic Operads: An Algorithmic Companion
Corollary 5.3.3.3. Let M be a symmetric collection. Then we have an iso-
morphism of shuffle operads
T
X
(M
f
)
∼
=
(T
Σ
(M))
f
.
Moreover, if I ⊂ T
Σ
(M) is an ideal, then, under the identification that we
made, I
f
is an ideal of T
X
(M
f
), and
T
X
(M
f
)/I
f
∼
=
(T
Σ
(M)/I)
f
.
Proof. All the notions in question, that is free symmetric operads, free shuffle
operads, and ideals in those operads, are defined using composition products,
so Proposition 5.3.3.1 applies.
Example 5.3.3.4. Let us consider the free shuffle operad T
X
(X) from Ex-
ample 5.3.1.3. Note that it is isomorphic to T
Σ
(L)
f
, where L is the symmetric
collection from Example 5.2.2.4: ...