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
Previous Page Next Page