Symmetric Monoidal Categories for Operads - Index of

15.2 Homotopy Invariance of Functors for Cofibrant Algebras 227
j ⊗ Fr ◦ R : C ⊗ Fr ◦ R → D ⊗ Fr ◦ R
since the functor (M,A) ↦→ SR(M,A) preserves colimits in M.
Consider the more general case of a morphism of free right R-modules
k ◦ R : K ◦ R → L ◦ R
induced by a cofibration (an acyclic cofibration) of Σ∗-objects k : K → L. In
the case of a right R-module of the form M = K ◦ R, wehave
SR(M,A) =SR(K ◦ R,A)=S(K, A)
by theorem 7.1.1. Therefore we obtain that the pushout-product
(i∗,f∗) :SR(M,B) �
SR(N,A) → SR(N,B)
SR(M,A)
is identified with the pushout-product
(k∗,f∗) :S(K, B) �
S(K,A)
S(L, A) → S(L, B)
of lemma 11.5.1. Thus we can deduce lemma 15.1.1 from the assertion of
lemma 11.5.1. ⊓⊔
As a corollary, we obtain:
15.1.2 Lemma.
(a) Let M be a cofibrant object in M R. If a morphism of R-algebras f : A →
B, whereAis E-cofibrant, forms a E-cofibration (respectively, an acyclic Ecofibration),
then the induced morphism SR(M,f) :SR(M,A) → SR(M,B)
defines a cofibration (respectively, an acyclic cofibration) in E.
(b) Let A be an E-cofibrant R-algebra. The morphism SR(i, A) :SR(M,A) →
SR(N,A) induced by a cofibration (respectively, an acyclic cofibration) of right
R-modules i : M → N forms a cofibration (respectively, an acyclic cofibration)
in E. ⊓⊔
And the claim of theorem 15.1.A follows immediately from Brown’s lemma.
⊓⊔
15.2 Homotopy Invariance of Functors for Cofibrant
Algebras
In this chapter, we study the homotopy invariance of the functors SR(M) :
RE →Eassociated to right R-modules M which are not cofibrant. Our main
result reads: