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The main purpose of the present work is to present to the reader a
particularly nice category for the study of homotopy, namely the
homo topic category (IV). This category is, in fact, - according to
Chapter VII and a well-known theorem of J. H. C. WHITEHEAD -
equivalent to the category of CW-complexes modulo homotopy, i.e.
the category whose objects are spaces of the homotopy type of a
CW-complex and whose morphisms are homotopy classes of continuous
mappings between such spaces. It is also equivalent (I, 1.3) to a
category of fractions of the category of topological spaces modulo
homotopy, and to the category of Kan complexes modulo homotopy
(IV). In order to define our homotopic category, it appears useful
to follow as closely as possible methods which have proved
efficacious in homo logical algebra. Our category is thus the"
topological" analogue of the derived category of an abelian
category (VERDIER). The algebraic machinery upon which this work is
essentially based includes the usual grounding in category theory -
summarized in the Dictionary - and the theory of categories of
fractions which forms the subject of the first chapter of the book.
The merely topological machinery reduces to a few properties of
Kelley spaces (Chapters I and III). The starting point of our study
is the category,10 Iff of simplicial sets (C.S.S. complexes or
semi-simplicial sets in a former terminology)."
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