Symmetric Monoidal Categories for Operads - Index of

170 11 Symmetric Monoidal Model Categories for Operads
defines a weak-equivalence for any cofibrant R-module E.Thisassertionisimmediate
for a free R-module E = R⊗C,becausewehaveS⊗R(R⊗C) =S⊗C
and η(R ⊗ C) is identified with the morphism ψ ⊗ C : R ⊗ C → S ⊗ C. The
assertion can easily be generalized to cofibrant cell R-modules (by using an
induction on the cell decomposition), because the functors U : R Mod →
dg k Mod and U(ψ ∗ ψ!−) : R Mod → dg k Mod map cell attachments in
R-module to pushouts in dg-modules. Use that any cofibrant R-module is
a retract of a cofibrant cell R-module to conclude.
Let F be any left S-module. Let E ∼ −→ ψ ∗ F be a cofibrant replacement of
ψ ∗ F in the category of left R-modules. Use the commutative diagram
η(E)
E
to prove that the composite
∼
η(ψ ∗ (F ))
ψ ∗ (F )
ψ ∗ ψ!(E) ψ ∗ ψ!ψ ∗ (F )
ψ ∗ (ɛ(F ))
ψ ∗ (F )
ψ!(E) → ψ!ψ ∗ (F )
ɛ(F )
−−−→ F
is a weak-equivalence if η(E) :E → ψ ∗ ψ!(E) isso(see§16.1.5).
From these verifications, we conclude that the pair (ψ!,ψ ∗ ) forms a Quillen
equivalence if ψ is a weak-equivalence. ⊓⊔
11.3 Symmetric Monoidal Model Categories over a Base
In the context of operads, we use model categories equipped with a symmetric
monoidal structure. With a view to applications in homotopy theory, we have
to put appropriate conditions on the tensor product and to introduce a suitable
notion of a symmetric monoidal model category. Good axioms are formalized
in [28, §§4.1-4.2]. The purpose of this section is to review these axioms
in the relative context of symmetric monoidal categories over a base and to
specify which axioms are really necessary for our needs.
11.3.1 The Pushout-Product. The pushout-product gives a way to assemble
tensor products of (acyclic) cofibrations in symmetric monoidal model
categories. The definition of the pushout-product makes sense for any bifunctor
T : A×B →X where X is a category with colimits. The pushout-product
of morphisms f : A → B and g : C → D is just the morphism
id