1.1. THE DEFINITION OF A MODEL CATEGORY

3

property with respect to i if, for every commutative diagram of the following form,

A —f—+ x

4

iP

B

•

Y

9

there is a lift h: B — * X such that hi = f and ph — g.

DEFINITION

1.1.3. A model structure on a category C is three subcategories of

C called weak equivalences, cofibrations, and fibrations, and two functorial factor-

izations (a, (3) and (7,6) satisfying the following properties:

1. (2-out-of-3) If / and g are morphisms of C such that gf is defined and two

of fyg and gf are weak equivalences, then so is the third.

2. (Retracts) If / and g are morphisms of C such that / is a retract of g and g

is a weak equivalence, cofibration, or fibration, then so is / .

3. (Lifting) Define a map to be a trivial cofibration if it is both a cofibration

and a weak equivalence. Similarly, define a map to be a trivial fibration if

it is both a fibration and a weak equivalence. Then trivial cofibrations have

the left lifting property with respect to fibrations, and cofibrations have the

left lifting property with respect to trivial fibrations.

4. (Factorization) For any morphism / , a(f) is a cofibration, /3(f) is a trivial

fibration, j(f) is a trivial cofibration, and 6(f) is a fibration.

DEFINITION

1.1.4. A model category is a category C with all small limits and

colimits together with a model structure on 6.

Recall that a category is said to have all small (co)limits if, for every small

category 3 (i.e. ob J, and hence also mor 3, are sets) and every functor F: 3 — + C,

a (co)limit of F exists in C.

This definition of a model category differs from the definition in [Qui67] in

the following ways. Recall that Quillen distinguished between model categories

and closed model categories. That distinction has not proved to be important, so

recent authors have only considered closed model categories. We therefore drop

the adjective closed. In addition, Quillen only required finite limits and colimits to

exist. All of the examples he considered where only such colimits and limits exist

are full subcategories of model categories where all small colimits and limits exist.

Since it is technically much more convenient to assume all small colimits and limits

exist, we do so. Quillen also assumed the factorizations merely exist, not that they

are functorial. However, in all the examples they can be made functorial.

The changes we have discussed so far are due to Kan and appear in [DHK].

We make one further change in that we make the functorial factorizations part

of the model structure, rather than merely assuming they exist. This is a subtle

difference, necessary for various constructions to be natural with respect to maps

of model categories.

We always abuse notation and refer to a model category C, leaving the model

structure implicit. We will discuss several examples of model categories in the next

two chapters. We can give some trivial examples now.

EXAMPLE

1.1.5. Suppose 6 is a category with all small colimits and limits. We

can put three different model structures on C by choosing one of the distinguished