Symmetric Monoidal Categories for Operads - Index of

138 9 Algebras in Right Modules over Operads
In §7.2.10, we record that
M R
SR
F R
ψ!
ψ ∗
ψ!
ψ ∗
M S
forms a diagram of functors of symmetric monoidal categories over C that
commutes up to a natural equivalence of symmetric monoidal categories over
C. As a consequence, by naturality of our constructions, we obtain:
SS
F S
9.4.5 Proposition. Let ψ : R → S be an operad morphism.
(a) The diagram
End M
ψ!
End ψ!M
ΘR
ΘS End SS(ψ!M)
commutes, for every right R-module M.
(b) The diagram
End N
ψ ∗
End ψ ∗ N
ΘS
ΘR End SR(ψ ∗ N)
End SR(M)
ψ!
� End ψ! SR(M) .
End SS(N)
ψ ∗
� End ψ ∗ SS(N) .
commutes, for every right S-module N. ⊓⊔