one in that we do not assume our chain complexes are bounded below. We then
move on to topological spaces. Here our treatment is the standard one, except that
we offer more details than are commonly provided. The model category of chain
complexes of comodules over a commutative Hopf algebra, on the other hand, has
not been considered before. It is relevant to the recent work in modular represen-
tation theory of Benson, Carlson, Rickard and others (see, for example [BCR96]),
as well as to the study of stable homotopy over the Steenrod algebra [Pal97]. The
approach to simplicial sets given in the third chapter is substantially the same as
that of [GJ97].
In the fourth chapter we consider model categories that have an internal tensor
product making them into closed monoidal categories. Almost all the standard
model categories are like this: chain complexes of abelian groups have the tensor
product, for example. Of course, one must require the tensor product and the model
structure to be compatible in an appropriate sense. The resulting monoidal model
categories play the same role in the theory of model categories that ordinary rings
do in algebra, so that one can consider modules and algebras over them. A module
over the monoidal model category of simplicial sets, for example, is the same thing
as a simplicial model category. Of course, the homotopy category of a monoidal
model category is a closed monoidal category in a natural way, and similarly for
modules and algebras. The material in this chapter is all fairly straightforward,
but has not appeared in print before. It may also be in [DHK], when that book
The fifth and sixth chapters form the technical heart of the book. In the fifth
chapter, we show that the homotopy category of any model category has the same
good properties as the homotopy category of a simplicial model category. In our
highfalutin language, the homotopy pseudo-2-functor lifts to a pseudo-2-functor
from model categories to closed Ho SSet-modules, where HoSSet is the homo-
topy category of simplicial sets. This follows from the idea of framings developed
in [DK80]. This chapter thus has a lot of overlap with [DHK], where framings are
also considered. However, the emphasis in [DHK] seems to be on using framings to
develop the theory of homotopy colimits and homotopy limits, whereas we are more
interested in making Ho SSet act naturally on the homotopy category. There is a
nagging question left unsolved in this chapter, however. We find that the homotopy
category of a monoidal model category is naturally a closed algebra over HoSSet,
but we are unable to prove that it is a central closed algebra.
In the sixth chapter we consider the homotopy category of a pointed model
category. As was originally pointed out by Quillen [Qui67], the apparently minor
condition that the initial and terminal objects coincide in a model category has
profound implications in the homotopy category. One gets a suspension and loop
functor and cofiber and fiber sequences. In the light of the fifth chapter, however,
we realize we get an entire closed HoSSet*-action, of which the suspension and
loop functors are merely specializations. Here Ho SSet* is the homotopy category
of pointed simplicial sets. We prove that the cofiber and fiber sequences are com-
patible with this action in an appropriate sense, as well as reproving the standard
facts about cofiber and fiber sequences. We then get a notion of pre-triangulated
categories, which are closed Ho SSet*-modules with cofiber and fiber sequences
satisfying many axioms.
The seventh chapter is devoted to the stable situation. We define a pre-
triangulated category to be triangulated if the suspension functor is an equivalence
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