Model categories, first introduced by Quillen in [Qui67], form the foundation of
homotopy theory. The basic problem that model categories solve is the following.
Given a category, one often has certain maps (weak equivalences) that are not
isomorphisms, but one would like to consider them to be isomorphisms. One can
always formally invert the weak equivalences, but in this case one loses control of
the morphisms in the quotient category. The morphisms between two objects in
the quotient category may not even be a set. If the weak equivalences are part of
a model structure, however, then the morphisms in the quotient category from X
to Y are simply homotopy classes of maps from a cofibrant replacement of X to a
fibrant replacement of Y.
Because this idea of inverting weak equivalences is so central in mathematics,
model categories are extremely important. However, so far their utility has been
mostly confined to areas historically associated with algebraic topology, such as
homological algebra, algebraic if-theory, and algebraic topology itself. The author
is certain that this list will be expanded to cover other areas of mathematics in
the near future. For example, Voevodksy's work [Voe97] is certain to make model
categories part of every algebraic geometer's toolkit.
These examples should make it clear that model categories are really funda-
mental. However, there is no systematic study of model categories in the literature.
Nowhere can the author find a definition of the category of model categories, for
example. Yet one of the main lessons of twentieth century mathematics is that to
study a structure, one must also study the maps that preserve that structure.
In addition, there is no excellent source for information about model categories.
The standard reference [Qui67] is difficult to read, because there is no index and
because the definitions are not ideal (they were changed later in [Qui69]). There
is also [BK72, Part II], which is very good at what it does, but whose emphasis is
only on simplicial sets. More recently, there is the expository paper [DS95], which
is highly recommended as an introduction. But there is no mention of simplicial
sets in that paper, and it does not go very far into the theory.
The time seems to be right for a more careful study of model categories from
the ground up. Both of the books [DHK] and [Hir97], unfinished as the author
writes this, will do this from different perspectives. The book [DHK] overlaps
considerably with this one, but concentrates more on homotopy colimits and less
on the relationship between a model category and its homotopy category. The
book [Hir97] is concerned with localization of model categories, but also contains
a significant amount of general theory. There is also the book [GJ97], which con-
centrates on simplicial examples. All three of these books are highly recommended
to the reader.
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