Introduction

Model categories and their homotopy categories

A model category is Quillen's axiomatization of a place in which you can "do

homotopy theory" [52]. Homotopy theory often involves treating homotopic maps

as though they were the same map, but a homotopy relation on maps is not the

starting point for abstract homotopy theory. Instead, homotopy theory comes from

choosing a class of maps, called weak equivalences, and studying the passage to

the homotopy category, which is the category obtained by localizing with respect

to the weak equivalences, i.e., by making the weak equivalences into isomorphisms

(see Definition 8.3.2). A model category is a category together with a class of

maps called weak equivalences plus two other classes of maps (called cofibrations

and fibrations) satisfying five axioms (see Definition 7.1.3). The cofibrations and

fibrations of a model category allow for lifting and extending maps as needed to

study the passage to the homotopy category.

The homotopy category of a model category. Homotopy theory origi-

nated in the category of topological spaces, which has unusually good technical

properties. In this category, the homotopy relation on the set of maps between two

objects is always an equivalence relation, and composition of homotopy classes is

well defined. In the classical homotopy theory of topological spaces, the passage

to the homotopy category was often described as "replacing maps with homotopy

classes of maps". Most work was with CW-complexes, though, and whenever a

construction led to a space that was not a CW-complex the space was replaced by

a weakly equivalent one that was. Thus, weakly equivalent spaces were recognized

as somehow "equivalent", even if that equivalence was never made explicit. If in-

stead of starting with a homotopy relation we explicitly cause weak equivalences

to become isomorphisms, then homotopic maps do become the same map (see

Lemma 8.3.4) and in addition a cell complex weakly equivalent to a space becomes

isomorphic to that space, which would not be true if we were simply replacing maps

with homotopy classes of maps.

In most model categories, the homotopy relation does not have the good prop-

erties that it has in the category of topological spaces unless you restrict yourself

to the subcategory of cofibrant-fibrant objects (see Definition 7.1.5). There are ac-

tually two different homotopy relations on the set of maps between two objects X

and Y: Left homotopy, defined using cylinder objects for X, and right homotopy,

defined using path objects for Y (see Definition 7.3.2). For arbitrary objects X

and Y these are different relations, and neither of them is an equivalence relation.

However, for cofibrant-fibrant objects, the two homotopy relations are the same,

they are equivalence relations, and composition of homotopy classes is well defined

(see Theorem 7.4.9 and Theorem 7.5.5). Every object of a model category is weakly

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