Symmetric Monoidal Categories for Operads - Index of

232 15 Modules and Homotopy Invariance of Functors
(b) Let i : M → N be any morphism of connected right R-modules. Let
f : A → B be a relative L-cell complex in the category of connected left
R-modules. Suppose A is an L-cell complex. If i is an (acyclic) C-cofibration,
then so is the pushout-product
(i∗,f∗) :M ◦R B �
N ◦R A → N ◦R B
induced by i.
M◦RA
Proof. These assertions improve on results of §§15.1-15.2 in the case where
we consider R-algebras in the category of connected Σ∗-objects M 0 R. Assertion
(a) corresponds to the case where i is a cofibration and f is an (acyclic)
C-cofibration in lemma 15.1.1 and is proved in by the same argument. Assertion
(b) arises from the extension of lemma 15.2.C to reduced categories with
regular tensor powers, in the case where i is an (acyclic) C-cofibration and
f is a cofibration between cofibrant objects. To prove this latter assertion,
check indications given in §15.2. ⊓⊔
As a corollary, we obtain:
15.3.2 Lemma.
(a) If M is a K-cell complex in the category of connected right R-modules,
then the functor M ◦R − maps (acyclic) C-cofibrations between C-cofibrant
objects in the category of connected left R-modules to weak-equivalences.
(b) If A is an L-cell complex in the category of connected left R-modules, then
the functor −◦R A maps (acyclic) C-cofibrations in the category of connected
right R-modules to weak-equivalences. ⊓⊔
And Brown’s lemma implies again:
15.3.3 Lemma.
(a) If M is a K-cell complex in the category of connected right R-modules,
then the functor M ◦R − maps weak-equivalences between C-cofibrant objects
in R M 0 to weak-equivalences.
(b) If A is an L-cell complex in the category of connected left R-modules, then
the functor −◦RA maps weak-equivalences between C-cofibrant objects in M 0 R
to weak-equivalences. ⊓⊔
Observe simply that K-cell (respectively, L-cell) complexes are C-cofibrant
and check the argument of [28, Lemma 1.1.12].
Obviously, we can extend this lemma to the case where M (respectively,
A) isaretractofaK-cell complex (respectively, of an L-cell complex).
Thus lemma 15.3.3 implies assertion (b) of theorem 15.3.A and assertion
(a) of theorem 15.3.B.
To achieve the proof of theorem 15.3.A and theorem 15.3.B, we observe: