Symmetric Monoidal Categories for Operads - Index of

168 11 Symmetric Monoidal Model Categories for Operads
11.2.6 Proposition. Suppose R is cofibrant as a dg-module.
The category of left R-modules inherits a cofibrantly generated model
structure so that the forgetful functor U : R Mod → dg k Mod creates
weak-equivalences and where the generating cofibrations (respectively, acyclic
cofibrations) are the morphisms of free left R-modules R ⊗ i : R ⊗ C → R ⊗ D
induced by generating cofibrations (respectively, acyclic cofibrations) of dgmodules.
The forgetful functor U : R Mod → dg k Mod creates fibrations as well and
preserves cofibrations.
Symmetric assertions hold for the category of right R-modules.
If R is not cofibrant as a dg-module, then the conditions of
proposition 11.1.14 are not satisfied. Nevertheless, the assumptions of theorem
11.1.13 can be checked directly without using the assumption of
proposition 11.2.6. Thus the category of left (respectively, right) R-modules
has still a cofibrantly generated model structure if R is not cofibrant as
a dg-module, but the forgetful functor U : R Mod → dg k Mod does not
preserve cofibrations.
Proof. Condition (1) of proposition 11.1.14 is satisfied since the forgetful
functor U : R Mod → dg k Mod creates all colimits. Condition (2) is implied
by the next observation, because the forgetful functor U : R Mod → dg k Mod
creates all colimits (including pushouts) and cofibrations (respectively, acyclic
cofibrations) are stable under pushouts in dg-modules. ⊓⊔
11.2.7 Fact. Let K be any cofibrant dg-module. If i is a cofibration (respectively,
an acyclic cofibration) in the category of dg-modules, then so is the
tensor product K ⊗ i : K ⊗ C → K ⊗ D.
This fact holds in the context of monoidal model categories as a consequence
of the pushout-product axiom, recalled next. But, in the context
of dg-modules, we can also use the explicit form of cofibrations yielded by
proposition 11.2.3 to check fact 11.2.7 directly.
11.2.8 Twisting Cochains and Quasi-Free Modules over dg-
Algebras. The constructions and results of §§11.2.1-11.2.4 have obvious generalizations
in the context of modules over dg-algebras. For a left R-module
E, we assume that a twisting cochain ∂ : E → E commutes with the R-action
to ensure that the map δ+∂ : E → E defines a differential of R-modules. Naturally,
we say that a left R-module E is quasi-free if we have E =(R⊗C, ∂),
where ∂ : R ⊗ C → R ⊗ C is a twisting cochain of left R-modules. In this
case, the commutation with the R-action implies that the twisting cochain
∂ : R ⊗ C → R ⊗ C is determined by a homogeneous map ∂ : C → R ⊗ C.
The extension of proposition 11.2.9 to modules over dg-algebras reads:
11.2.9 Proposition. A cofibrant cell object in left R-modules is equivalent
toaquasi-freeR-module L =(R ⊗ K, ∂), whereK is a free graded k-module