This monograph, which is aimed at the graduate level and beyond, consists of
two parts.
In part II we develop the beginnings of a kind of “relative” category theory of
what we will call homotopical categories. These are categories with a single distinguished
class of maps (called weak equivalences) containing all the isomorphisms
and satisfying one simple two out of six axiom which states that
(*) for every three maps r, s and t for which the two compositions sr and ts
are defined and are weak equivalences, the four maps r, s, t and tsr are
also weak equivalences,
which enables one to define “homotopical” versions of such basic categorical notions
as initial and terminal objects, colimit and limit functors, adjunctions, Kan
extensions and universal properties.
In part I we then use the results of part II to get a better understanding of
Quillen’s so useful model categories, which are categories with three distinguished
classes of maps (called cofibrations, fibrations and weak equivalences) satisfying a
few simple axioms which enable one to “do homotopy theory”, and in particular to
show that such model categories are homotopically cocomplete and homotopically
complete in a sense which is much stronger than the existence of all small homotopy
colimit and limit functors.
Both parts are essentially self-contained. A reader of part II is assumed to have
some familiarity with the categorical notions mentioned above, while those who read
part I (and especially the introductory chapter) should also know something about
model categories. In the hope of increasing the local as well as the global readability
of this monograph, we not only start each section with some introductory remarks
and each chapter with an introductory section, but also each of the two parts with
an introductory chapter, with the first chapter of part I as motivation for and
introduction to the whole monograph and the first chapter of part II summarizing
the main results of its other three chapters.