Symmetric Monoidal Categories for Operads - Index of

82 4 Miscellaneous Structures Associated to Algebras over Operads
and we can use the natural map
ρ :Mor P / OE (P[X], S) → MorE(X, S(0))
which associates to any φ : P[X] → S the composite
X η[X]
−−−→ P[X](0) φ −→ S(0).
Observe that the canonical morphism
P(m + n) ⊗ X ⊗n → (P(m + n) ⊗ X ⊗n )Σn ↩→ S(P[m],X)=P[X](m)
is identified with the composite of the operadic composition product
with the morphism
P[X](m + n) ⊗ P[X](1) ⊗m ⊗ P[X](0) ⊗n → P[X](m)
P(m + n) ⊗ 1 ⊗m ⊗X ⊗n → P[X](m + n) ⊗ P[X](1) ⊗m ⊗ P[X](0) ⊗n
induced by the operad morphism η : P → P[X], the operad unit η : 1 →
P[X](1) and η[X] :X → P[X](0). Consequently, any morphism φ : P[X] → S
of operads under P fits a commutative diagram of the form
P(m + n) ⊗ X ⊗n
�
P(m + n) ⊗ 1 ⊗m ⊗X ⊗n
η⊗η ⊗m ⊗f ⊗n
S(m + n) ⊗ S(1) ⊗m ⊗ S(0) ⊗n
ν
P[X](m)
φ
S(m),
where f = ρ(φ) andν refers to the composition product of S. Thusweobtain
that φ : P[X] → S is determined by the associated morphism f = ρ(φ).
If we are given a morphism f : X → S(0), then straightforward verifications
show that the morphisms φ : P[X](m) → S(m) determined by the
diagram define a morphism of operads under P.
Hence we conclude that the map φ ↦→ ρ(φ) is one-to-one. ⊓⊔
In the case S = P[Y ], we obtain that any morphism of P-algebras f :
P(X) → P(Y ) determines a morphism φf : P[X] → P[Y ] in the category of
operads under P. The definition of the correspondence
Mor P / OE (R, S) → Mor PE(R(0), S(0))