Symmetric Monoidal Categories for Operads - Index of

11.1 Recollections: The Language of Model Categories 163
object argument holds for F I (respectively, F J ) and this achieves the proof
of condition (1) of theorem 11.1.13.
Our first observation implies that the functor U : A→X maps relative
F I-cell complexes to cofibrations. As a byproduct, we obtain readily that
U : A→X preserves cofibrations since cofibrations are retracts of relative
F I-cell complexes. ⊓⊔
11.1.15 Recollections on Quillen’s Adjunctions. Nowwesupposegiven
an adjunction F : X ⇄ A : U between well-defined model categories A and X .
In accordance with usual conventions, we say that the functors (F, U) define
a Quillen adjunction if any one of the following equivalent condition holds
(we refer to [27, §8.5] and [28, §1.3]):
A1. The functor F preserves cofibrations and acyclic cofibrations.
A2. The functor U preserves fibrations and acyclic fibrations.
A3. The functor F preserves cofibrations and U preserves fibrations.
The functors (F, U) define a Quillen equivalence if we have further:
E1. For every cofibrant object X ∈X,thecomposite
X ηX
−−→ UF(X) → U(B)
forms a weak-equivalence in X ,whereηX refers to the adjunction unit
and B is any fibrant replacement of F (X).
E2. For every fibrant object A ∈A,thecomposite
F (Y ) → FU(A) ɛA
−→ A
forms a weak-equivalence in A, whereɛA refers to the adjunction augmentation
and Y is any cofibrant replacement of U(A).
The notion of a Quillen adjunction gives usual conditions to define adjoint
derived functors on homotopy categories. The notion of a Quillen equivalence
gives a usual condition to obtain adjoint equivalences of homotopy categories.
In this book, we define examples of Quillen adjunctions (respectively, equivalences),
but we do not address applications to the construction of derived
functors for which we refer to the literature.
We use essentially:
(a) In a Quillen adjunction, the left adjoint F : X → A preserves weakequivalences
between cofibrant objects. This assertion is a consequence of
the so-called Brown’s lemma, recalled next in a generalized context.
(b) In a Quillen equivalence, the adjunction relation maps a weakequivalence
φ : X ∼ −→ U(B) sothatX is cofibrant and B is fibrant to a
weak-equivalence φ♯ : F (X) ∼ −→ B.
11.1.16 Enlarged Classes of Cofibrations. In certain constructions, we
have functors F : A→X which preserve more (acyclic) cofibrations than the