6 I. MODEL APPROXIMATIONS AND BOUNDED DIAGRAMS

For a detailed exposition on colimits and limits we refer the reader to [31, 39].

The colimit (respectively the limit) is a particular example of a more general left

(respectively right) Kan extension. For these and other categorical constructions

used in this paper see Appendix B.

Let C be a category and W a class of morphisms in C. We say that W satisfies

the "two out of three" property when for any composable morphisms / : X — Y

and g : Y — Z in C, if two out of / , g, and gof belong to W, then so does the third.

A category with weak equivalences is by definition a category with a distinguished

class of morphisms that contains all isomorphisms and satisfies the "two out of

three" property. We use the symbol "-^i" to denote a morphism in this class.

Let C be a category with weak equivalences. A functor C —• Ho(C) is called

the localization of C with respect to weak equivalences if it satisfies the following

universal property:

• weak equivalences in C are sent via C —• Ho{C) to isomorphisms (this functor

is homotopy invariant);

• if C — £ is another functor which sends weak equivalences to isomorphisms,

then it can be expressed uniquely as a composite C — Ho(C) — * £ (where

C — » Ho(C) is the localization).

We say that a category with weak equivalences C admits a localization if the

functor C — » Ho(C) exists.

Let C be a category with weak equivalences and / be a small category. Let

^ : F — » G be a natural transformation between functors F : / — » C and G : I — C.

We say that ^ is a weak equivalence if it is an objectwise weak equivalence, i.e., if

for any i G I, ^ : F(i) — » G(i) is a weak equivalence in C. In this way Fun(I,C)

becomes a category with weak equivalences.

2. Mode l categories

In this section we review classical homotopical properties of the coproduct,

push-out, and the sequential colimit constructions. In order to be able to consider

their homotopical properties, we look at these constructions in model categories,

i.e., in categories in which one can do homotopy theory. We refer the reader to [35,

31, 41 , 42] for the necessary definitions and theorems concerning these categories.

Here we just sketch some of their properties. However the reader should keep in

mind that the notion of a model category is essential in this exposition; in fact this

paper is about model categories.

A model category is a category, which we usually denote by A4, together with

three distinguished classes of morphisms: weak equivalences , fibrations, and cofi-

brations . This structure is subject to five axioms M C 1 - M C 5 (see [31, Section

3]). A morphism which is both a weak equivalence and a fibration (respectively a

cofibration) is called an acyclic fibration (respectively an acyclic cofibration ). To

denote a weak equivalence, a cofibration, and a fibration we use respectively the

symbols «^","c-»? and "-»".

Axiom M C I guarantees that model categories are equipped with arbitrary

colimits and limits. In particular there is a terminal object, denoted by *, as

well as an initial object, denoted by 0. An object A is said to be cofibrant if the

morphism 0 — » A is a cofibration. It is said to be fibrant if the morphism A —• *

is a fibration. This axiom also implies the existence of products and coproducts in

M, denoted respectively by the symbols "17" and "]J".