Symmetric Monoidal Categories for Operads - Index of

228 15 Modules and Homotopy Invariance of Functors
Theorem 15.2.A. Let R be an operad. Suppose that R is Σ∗-cofibrant so that
the category of R-algebras is equipped with a semi-model structure.
(a) If A is cofibrant in the category of R-algebras, then the morphism
SR(f,A) :SR(M,A) → SR(N,A)
induced by a weak-equivalence of right R-modules f : M ∼ −→ N forms a weakequivalence
as long as M,N are Σ∗-cofibrant.
(b) Let M be a Σ∗-cofibrant right R-module. The morphism
SR(M,f) :SR(M,A) → SR(M,B)
induced by a weak-equivalence of R-algebras f : A ∼ −→ B forms a weakequivalence
if A and B are cofibrant in the category of R-algebras.
¶ Remark. In the context of a (reduced) symmetric monoidal category
with regular tensor powers, we assume that R is a C-cofibrant non-unitary
operad. Assertion (a) holds for any weak-equivalence of connected right
R-modules f : M ∼ −→ N such that M,N are C-cofibrant. Assertion (b) holds
for all connected right R-modules M which are C-cofibrant.
¶ Remark. Suppose E = C =Sp Σ is the category of symmetric spectra
together with the stable positive model structure. Let R be an operad
in simplicial sets. If we can ensure that the forgetful functor U : RE → E
preserves cofibrations, then our arguments work for right R-modules which
are cofibrant in the category of spectra. Hence:
– assertion (a) of theorem 15.2.A holds for any weak-equivalence of right
R-modules in spectra f : M ∼ −→ N such that M,N are Sp Σ -cofibrant;
– assertion (b) of theorem 15.2.A holds for all right R-modules in spectra M
which are Sp Σ -cofibrant.
Observe that assertion (a) of theorem 15.2.A has a converse:
Proposition 15.2.B. Assume E = C.
Let f : M → N be a morphism of right R-modules, where M,N are Σ∗cofibrant.
If the morphism SR(f,A) : SR(M,A) → SR(N,A) induced by f
forms a weak-equivalence in C for every cofibrant R-algebra A ∈ RE, thenf is
a weak-equivalence as well.
If C is pointed, then this implication holds without the assumption that the
modules M,N are Σ∗-cofibrant.
Proof. In proposition 11.5.4, we prove a similar assertion for the bifunctor
(M,X) ↦→ S(M,X), where M ∈Mand X ∈C. Proposition 15.2.B is an immediate
corollary of this result because the free R-algebras A = R(X), where
X is a cofibrant object in C, are cofibrant R-algebras and, by theorem 7.1.1,
we have SR(M,R(X)) � S(M,X) forafreeR-algebra A = R(X). ⊓⊔