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 Y of C that takes v to w.
Note that C* has arbitrary limits and colimits. Indeed, if F: 3 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 » C,
where G(*) = *, and G of the map * * 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 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* C, and defines a faithful (but not full) embedding of
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