Calculus of Fractions and Homotopy Theory

  • Peter Gabriel
  • Michel Zisman

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 35)

Table of contents

  1. Front Matter
    Pages II-X
  2. Peter Gabriel, Michel Zisman
    Pages 1-6
  3. Peter Gabriel, Michel Zisman
    Pages 6-21
  4. Peter Gabriel, Michel Zisman
    Pages 21-41
  5. Peter Gabriel, Michel Zisman
    Pages 41-56
  6. Peter Gabriel, Michel Zisman
    Pages 57-78
  7. Peter Gabriel, Michel Zisman
    Pages 78-106
  8. Peter Gabriel, Michel Zisman
    Pages 106-131
  9. Peter Gabriel, Michel Zisman
    Pages 131-139
  10. Back Matter
    Pages 139-168

About this book


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).


Calculus Homotopie Homotopy Morphism algebra category theory theorem

Authors and affiliations

  • Peter Gabriel
    • 1
  • Michel Zisman
    • 1
  1. 1.Departement de Mathématique StrasbourgUniversité de StrasbourgFrance

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin · Heidelberg 1967
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-85846-8
  • Online ISBN 978-3-642-85844-4
  • Buy this book on publisher's site