2

1. MODEL CATEGORIES

a morphism of model categories to be an adjoint pair, where the left adjoint is a

left Quillen functor and the right adjoint is a right Quillen functor. Of course, we

still have to pick a direction, but it is now immaterial which direction we pick. We

choose the direction of the left adjoint.

A Quillen functor will induce a functor on the homotopy categories, called its

total (left or right) derived functor. This operation of taking the derived functor

does not preserve identities or compositions, but it does do so up to coherent natural

isomorphism. We describe this precisely in Section 1.3, as is also done in [DHK]

but has never been done explicitly in print before that.

This observation leads naturally to 2-categories and pseudo-2-functors, which

we discuss in Section 1.4. The category of model categories is not really a category

at all, but a 2-category. The operation of taking the homotopy category and the

total derived functor is not a functor, but instead is a pseudo-2-functor. The 2-

morphisms of model categories are just natural transformations, so this section

really just points out that there is a convenient language to talk about these kind

of phenomena, rather than introducing any deep mathematics. This language is

convenient for the author, who will use it throughout the book. However, the reader

who prefers not to use it should skip this section and refer back to it as needed.

1.1. The definition of a model category

In this section we present our definition of a model category, and derive some

basic results. As mentioned above, our definition is different from the original

definition of Quillen and is even slightly different from the modern refinements

of [DHK]. The reader is thus advised to look at the definition we give here and

read the comments following it, even if she is familiar with model categories.

Other sources for model categories and basic results about them include the

original source [Qui67], the very readable [DS95], and the more modern [DHK]

and [Hir97]. Also, Robert Thomason [Wei97] was working on a different approach

to model categories just before he died.

We make some preliminary definitions.

Given a category C, we can form the category Map C whose objects are mor-

phisms of G and whose morphisms are commutative squares.

DEFINITION 1.1.1. Suppose 6 is a category.

1. A map / in C is a retract of a map g G C if / is a retract of g as objects

of Map 6. That is, / is a retract of g if and only if there is a commutative

diagram of the following form,

A C A

B • D B

where the horizontal composites are identities.

2. A functorial factorization is an ordered pair (a, /3) of functors Map C —

MapC such that / = /3(f) o a(f) for all / G Map 6. In particular, the

domain of a(f) is the domain of / , the codomain of a(f) is the domain of

/?(/), and the codomain of /3(f) is the codomain of / .

DEFINITION

1.1.2. Suppose i: A — B and p\ X — * Y are maps in a category

C. Then i has the left lifting property with respect to p and p has the right lifting