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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,