4 1. MODEL CATEGORIES

subcategories to be the isomorphisms and the other two to be all maps of 6. There

are then obvious choices for the functorial factorizations, and this gives a model

structure on C. For example, we could define a map to be a weak equivalence if

and only if it is an isomorphism, and define every map to be both a cofibration and

a fibration. In this case, we define the functors a and 6 to be the identity functor,

and define /3(f) to be the identity of the codomain of / and 7(/) to be the identity

of the domain of / .

EXAMPLE

1.1.6. Suppose 6 and V are model categories. Then 6 x D becomes

a model category in the obvious way: a map (/, g) is a cofibration (fibration, weak

equivalence) if and only if both / and g are cofibrations (fibrations, weak equiva-

lences). We leave it to the reader to define the functorial factorizations and verify

that the axioms hold. We could do this with any set of model categories. We refer

to the model structure just defined as the product model structure.

REMARK

1.1.7. A very useful property of the axioms for a model category is

that they are self-dual. That is, suppose 6 is a model category. Then the opposite

category Cop is also a model category, where the cofibrations of Cop are the fibrations

of C, the fibrations of Cop are the cofibrations of C, and the weak equivalences of

Cop

are the weak equivalences of C. The functorial factorizations also get inverted:

the functor a of

Cop

is the opposite of the functor 6 of 6, the functor (3 of

Cop

is the

opposite of the functor 7 of C, the functor 7 of

Cop

is the opposite of the functor

P of C, and the functor 6 of

Cop

is the opposite of the functor a of 6. We leave

it to the reader to check that these structures make

Cop

into a model category.

We denote it by DC, and refer to DC as the dual model category of C. Note that

D2C = C as model categories. In practice, this duality means that every theorem

about model categories has a dual theorem.

If 6 is a model category, then it has an initial object, the colimit of the empty

diagram, and a terminal object, the limit of the empty diagram. We call an object

of C cofibrant if the map from the initial object 0 to it is a cofibration, and we call

an object fibrant if the map from it to the terminal object 1 (or *) is a fibration. We

call a model category (or any category with an initial and terminal object) pointed

if the map from the initial object to the terminal object is an isomorphism.

Given a model category C, define C* to be the category under the terminal

object *. That is, an object of C* is a map * -^ X of C, often written (X, v). We

think of (X, v) as an object X together with a basepoint v. A morphism from (X, v)

to (F, w) is a morphism X â€” Y of C that takes v to w.

Note that C* has arbitrary limits and colimits. Indeed, if F: 3 â€” C* is a

functor from a small category 3 to C*, the limit of F as a functor to C is naturally

an element of C* and is the limit there. The colimit is a little trickier. For that, we

let 3 denote 3 with an extra initial object *. Then F defines a functor G: 3 â€” Â» C,

where G(*) = *, and G of the map * â€” * i is the basepoint of F(i). The colimit of

G in C then has a canonical basepoint, and this defines the colimit in C* of F. For

example, the initial object, the colimit of the empty diagram, in C* is *, and the

coproduct of X and Y is X V Y, the quotient of XIIY obtained by identifying the

basepoints. In particular, 6* is a pointed category.

There is an obvious functor 6 â€” C* that takes X to X+ = X U *, with

basepoint *. This operation of adding a disjoint basepoint is left adjoint to the

forgetful functor U: C* â€” C, and defines a faithful (but not full) embedding of