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1. MODEL CATEGORIES
The retract argument implies that the axioms for a model category are overde-
termined.
LEMMA
1.1.10. Suppose C is a model category. Then a map is a cofibration
(a trivial cofibration) if and only if it has the left lifting property with respect to
all trivial fibrations (fibrations). Dually, a map is a fibration (a trivial fibration)
if and only if it has the right lifting property with respect to all trivial cofibrations
(cofibrations).
PROOF. Certainly every cofibration does have the left lifting property with
respect to trivial fibrations. Conversely, suppose / has the left lifting property
with respect to trivial fibrations. Factor / = pi, where i is a cofibration and p
is a trivial fibration. Then / has the left lifting property with respect to p, so
the retract argument implies that / is a retract of i. Therefore / is a cofibration.
The trivial cofibration part of the lemma is proved similarly, and the fibration and
trivial fibration parts follow from duality.
In particular, every isomorphism in a model category is a trivial cofibration
and a trivial fibration, as is also clear from the retract axiom.
COROLLARY
1.1.11. Suppose e is a model category. Then cofibrations (trivial
cofibrations) are closed under pushouts. That is, if we have a pushout square as
follows,
A C
4 -I
B D
where f is a cofibration (trivial cofibration), then g is a cofibration (trivial cofibra-
tion). Dually, fibrations (trivial fibrations) are closed under pullbacks.
PROOF.
Because g is a pushout of / , if / has the left lifting property with
respect to a map h, so does g.
An extremely useful result about model categories is Ken Brown's Lemma.
LEMMA 1.1.12 (Ken Brown's lemma). Suppose C is a model category and T
is a category with a subcategory of weak equivalences that satisfies the two out of
three axiom. Suppose F: 6 D is a functor that takes trivial cofibrations betwen
cofibrant objects to weak equivalences. Then F takes all weak equivalences between
cofibrant objects to weak equivalences. Dually, if F takes trivial fibrations between
fibrant objects to weak equivalences, then F takes all weak equivalences between
fibrant objects to weak equivalences.
PROOF. Suppose / : A B is a weak equivalence of cofibrant objects. Factor
the map (/, 1#): AUB B into a cofibration A II B * C followed by a trivial
fibration C * B. The pushout diagram below
0 A
B AUB
shows that the inclusion maps A ^ AUB and similarly B ^ AUB are cofibra-
tions. By the two out of three axiom, both g o ^ and qoi2 are weak equivalences,
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