Symmetric Monoidal Categories for Operads - Index of

280 19 Shifted Functors and Pushout-Products
The pushout-product (k∗,λ∗) forms a cofibration in E if k and λ are so,
an acyclic cofibration if k or λ is also an acyclic cofibration. Since λ :
LnM[Y/X] → TnM[Y/X] is defined by an n-fold pushout-product in E,
we obtain that λ forms a cofibration (respectively, an acyclic cofibration) if
j is so. The conclusion follows. ⊓⊔
19.2 Pushouts
In this section, we study the morphism f : A → B obtained by a pushout
R(X)
R(j)
u
A
R(Y ) v B,
where R(j) :R(X) → R(Y ) is a morphism of free R-algebras induced by a
morphism j : X → Y in E. Our goal is to prove:
Lemma 19.2.A. Let A be an R-algebra. Suppose that the initial R-algebra
morphism η : R(0) → A forms a Σ∗-flat cofibration. If j is a cofibration
(respectively, an acyclic cofibration) in E, then the morphism f : A → B
obtained by a pushout of R(j) :R(X) → R(Y ) is a Σ∗-flat cofibration (respectively,
a Σ∗-flat acyclic cofibration).
We begin the proof of lemma 19.2.A by an observation:
19.2.1 Lemma. If η : R(0) → A is a Σ∗-flat cofibration, then the morphism
SR[i, A] :SR[M,A] → SR[N,A] forms a Σ∗-cofibration (respectively, an acyclic
Σ∗-cofibration) whenever i : M → N is so.
Proof. For the initial algebra R(0), proposition 18.2.1 gives an identity
SR[M,R(0)] = S[M,0] = M. Hence the pushout-product (i∗,η∗) isidentified
with the morphism SR[M,A] �
M N → SR[N,A] induced by SR[i, A] and
the natural morphism N → SR[N,A]. The pushout
M SR[M,A]
N
f
SR[M,A] �
M N
returns a Σ∗-cofibration (respectively, an acyclic Σ∗-cofibration) if i is so.
Hence, if η : R(0) → A is a Σ∗-flat cofibration, then we obtain that the
composite
SR[M,A] → SR[M,A] �
N → SR[N,A]
forms a Σ∗-cofibration (respectively, an acyclic Σ∗-cofibration) as well. ⊓⊔
M