Symmetric Monoidal Categories for Operads - Index of

15.3 Refinements for Relative Composition Products 231
induced by a weak-equivalence of connected right R-modules f : M ∼ −→ N
forms a weak-equivalence as long as M,N are retracts of K-cell complexes,
for every C-cofibrant object A ∈ R M 0 .
(b) If a connected right R-module M is a retract of a K-cell complex, then
the morphism
M ◦R g : M ◦R A → M ◦R B
induced by a weak-equivalence of connected left R-modules g : A ∼ −→ B forms
a weak-equivalence as long as A, B are C-cofibrant.
Theorem 15.3.B. Let R be a C-cofibrant non-unitary operad.
(a) If a connected left R-module A is a retract of an L-cell complex, then the
morphism
f ◦R A : M ◦R A → N ◦R A
induced by a weak-equivalence of connected right R-modules f : M ∼ −→ N
forms a weak-equivalence as long as M,N are C-cofibrant.
(b) The morphism
M ◦R g : M ◦R A → M ◦R B
induced by a weak-equivalence of connected left R-modules g : A ∼ −→ B forms
a weak-equivalence as long as A, B are retracts of L-cell complexes, for every
C-cofibrant object M ∈M 0 R.
One can observe further that theorem 15.3.A holds for any category of
algebras in a reduced category with regular tensor powers, and not only for
connected left R-modules, equivalent to algebras in the category of connected
Σ∗-objects.
We mention these results as remarks and we give simply a sketch of the
proof of theorem 15.3.A and theorem 15.3.B. In the case C = dgk Mod,
the category of dg-modules over a ring k, another proof can be obtained by
spectral sequence techniques (see the comparison theorems in [14, §2]).
The proof of theorem 15.3.A and theorem 15.3.B is based on the following
lemma:
15.3.1 Lemma.
(a) Let i : M → N be a relative K-cell complex in the category of connected
right R-modules. Let f : A → B be any morphism of connected left R-modules.
Suppose A is C-cofibrant. If f is an (acyclic) C-cofibration, then so is the
pushout-product
(i∗,f∗) :M ◦R B �
N ◦R A → N ◦R B
associated to i and f.
M◦RA