Calculus of Fractions and Homotopy Theory

1. Front Matter

   Pages II-X

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2. Dictionary

   • Peter Gabriel, Michel Zisman

   Pages 1-6

3. Categories of Fractions

   • Peter Gabriel, Michel Zisman

   Pages 6-21

4. Simplical Sets

   • Peter Gabriel, Michel Zisman

   Pages 21-41

5. Geometric Realization of Simplicial Sets

   • Peter Gabriel, Michel Zisman

   Pages 41-56

6. The Homotopic Category

   • Peter Gabriel, Michel Zisman

   Pages 57-78

7. Exact Sequences of Algebraic Topology

   • Peter Gabriel, Michel Zisman

   Pages 78-106

8. Exact Sequences of the Homotopic Category

   • Peter Gabriel, Michel Zisman

   Pages 106-131

9. Combinatorial Description of Topological Spaces

   • Peter Gabriel, Michel Zisman

   Pages 131-139

10. Back Matter

    Pages 139-168

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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).

• Calculus
• Homotopie
• Homotopy
• Morphism
• algebra
• category theory
• theorem

• Departement de Mathématique Strasbourg, Université de Strasbourg, France

  Peter Gabriel, Michel Zisman

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