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By Professor Dr. Peter Gabriel, Professor Dr. Michel Zisman (auth.)

The major goal of the current paintings is to offer to the reader a very great classification for the research of homotopy, specifically the homo­ subject class (IV). This type is, actually, - in accordance with bankruptcy VII and a widely known theorem of J. H. C. WHITEHEAD - corresponding to the class of CW-complexes modulo homotopy, i.e. the class whose gadgets are areas of the homotopy kind of a CW-complex and whose morphisms are homotopy sessions of constant mappings among such areas. it's also identical (I, 1.3) to a class of fractions of the class of topological areas modulo homotopy, and to the class of Kan complexes modulo homotopy (IV). with the intention to outline our homotopic classification, it seems that important to persist with as heavily as attainable equipment that have proved efficacious in homo­ logical algebra. Our type is hence the" topological" analogue of the derived type of an abelian type (VERDIER). The algebraic equipment upon which this paintings is largely dependent contains the standard grounding in classification conception - summarized within the Dictionary - and the speculation of different types of fractions which varieties the topic of the 1st bankruptcy of the e-book. The in basic terms topological equipment reduces to a couple houses of Kelley areas (Chapters I and III). the place to begin of our learn is the class ,10 Iff of simplicial units (C.S.S. complexes or semi-simplicial units in a former terminology).

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Extra info for Calculus of Fractions and Homotopy Theory

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Let us show that it is an injection: let fl' 'V: [m] - [n] be two morphlsms of Ll' with the same image in Ll ([m], en]). Since the relations (*) are verified in Ll', it is easily seen that fl and 'V have a decomposition in Ll' of the form (**). (The unicity of such a decomposition is not given a priori). The images of these decompositions in Ll are the canonical decomposition of fl and 'V. Since these images coincide, the decompositions of fl and 'V will also coincide, and hence fl='V. D. 3. e.

More generally, given Sk ,,-lX, we want to find Sk "X. Let 1:" be the set of non-degenerate n-simplices of X; let (LI [n ]a)aEX" be a family of standard n-simplices indexed by E"; and let ij: LI [n Ja-+X be the singular simplex associated with the non degenerate simplex a. We then have the following proposition: Proposition: With_the above notations, the square of Fig. 6. 6. it is sufficient to check that for each the square of Fig. 7 p;£ n, 4. Simplicial Sets and Category of Categories 31 is cocartesian.

1. 5. 2, let us now determine the functor I ? I. 9 P u li O~i

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