Table Of ContentErgebnisse der Mathematik
und ihrer Grenzgebiete
Band 35
Herausgegeben von
P. R. Halmos . P. J. Hilton· R. Remmert· B. Szokefalvi-Nagy
Unter Mitwirkung von
L. V. Ahlfors . R. Baer . F. L. Bauer· R. Courant· A. Dold . J. L. Doob
S. Eilenberg . M. Kneser . H. Rademacher' B. Segre . E. Sperner
Redaktion: P. J. Hilton
Calculus of Fractions
and Homotopy Theory
P. Gabriel and M. Zisman
Springer-Verlag New York Inc. 1967
Professor Dr. Peter Gabriel
Professor Dr. Michel Zisman
Universite de Strasbourg
Departement de Mathematique Strasbourg
ISBN 978-3-642-85846-8 ISBN 978-3-642-85844-4 (eBook)
DOl 10.1007/978-3-642-85844-4
All rights reserved, especially that of translation into foreign languages. It is also forbidden to reproduce
this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)
without written permission from the Publishers. © by Springer-Verlag Berlin . Heidelberg 1967.
Softcover reprint of the hardcover 1st edition 1967
Library of Congress Catalog Card Number 67-10470
Title-No. 4579
Introduction
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).
Notwithstanding a very large number of papers and seminar notes
published on the subject, no book has yet been devoted to them. Therefore
in order to fill this gap to some extent, we go back to the beginning of
the theory, and give a complete proof of theorems well-known to the
specialist, in the hope of providing the reader with a coherent survey,
and presenting some proofs which are easier or more conceptual than
those already published.
This book is thus intended to appeal at the same time to the beginner
who wishes to learn algebraic topology, to the algebraist who wants to
be acquainted with topology, and to the topologist eager to assimilate
the category language.
Such a program, which a priori seems very ambitious, is in fact very
limited: it cannot be greater than the number of 'pages in the volumes
where this work has been published. Thus the point where we leave off
VI In lroJ uction
is, in fad, nothing hilt the starting point of algehraic topology, and this
book is thus only an introduction to that theory.
Let us summarize briefly the content of our work.
Chapter I sets forth the theory of categories of fractions and gives
a few examples of applications to groupoids, Kelley spaces and abelian
categories. Given a category 'C and a subset 1: of the set dr 'C of the
morphisms of 'C, a category 'C [1:-1] is constructed whose objects are the
same as those of 'C, but where the morphisms of 1: have been formally
made invertible. The description of the set dr 'C[ 1:-1] is particularly
nice when 1: possesses some properties" allowing a calculus 0/ /ractions"
since in that case any morphism of 'C [1:-1] can be written S-l/ where s
is in 1: and / in dr 'C. The interest of this concept lies mainly in its
relationship to the existence of adjoint functors (I, 1.3 and 1,4.1).
After having recalled a few properties of the category of functors
with values in a set, Chapter II gives the definition of the category Llo tfj'
of simplicial sets, and draws the first inferences from it. One constructs
a fully faithful functor from the category 'Cat into Llo tfj' which has a left
adjoint. This pair of adjoint functors allows us to define certain other
pairs of adjoint functors, and in particular the pair (II, D) where IIX
is the Poincare groupoid of the simplicial set X, and DG is a K (III G, 1)
complex where III G is the Poincare group of the groupoid G. This concept
allows us finally to construct an extremely simple theory for the funda
mental group of a pointed simplicial set, and in particular to state a
Van Kampen theorem in the category . LI ° tfj'.
I I,
Chapter III is concerned with the study of the functor ? that is,
MILNOR'S geometric realization functor. After having shown that the
geometric realization of a simplicial set has some good properties (it is
a Hausdorff space, locally arc wise connected and locally contractible), it
I I
is proved that the functor ? has some interesting exactness properties
too, and that it commutes with locally trivial morphisms provided one
considers the range 0/ I ? I to be the category 0/ Kelley spaces instead of the
whole category of topological spaces. Under this new definition, the
geometric realization functor commutes with direct limits and finite
inverses limits. Moreover it transforms a locally trivial morphism into
a Serre fibration.
With Chapter IV the study of homotopy begins. After having defined
the homotopy relation between morphisms without any restriction (and
not only when the common range is a Kan complex), the category LID tfj'
of complexes modulo homotopy and a special set of arrows in that
category - the anodyne extensions -, it only remains for us to define
the homotopic category ;Yf as the category of fractions of Llo tfj' where the
anodyne extensions are made invertible. Since, for any simplicial set X,
Introduction VII
there exists an anodyne extension ax: X -+XK where XK is a Kan
complex, the category ;t is equivalent to the category of Kan complexes
modulo homotopy. Chapter IV also gives a variant "with base points"
of the preceding theory, and contains a few technical results on Kan
fibrations and Kan complexes. It is finally pointed out - as an
exercise - how the lI-theory given in Chapter II fits into this new
context.
Chapter V is independent of the preceding ones and presupposes
only a few elementary results (recalled in the Dictionary and in Chapter I)
about groupoids. Its purpose is to give a standard and self-dual proof of
various exact sequences occuring in algebraic topology. As an example
of possible applications, the reader will find the proof of a few well
known exact sequences (PUPPE, ECKMANN-HILTON); he will be able to
obtain in the same way all the other exact sequences of ECKMANN
HILTON [1]. The main idea is to construct an exact sequence in the
2-category of pointed groupoids and then to reduce the study of a large
class of 2-categories to the preceding one.
We should point out that the preceding method allows to give an
easy proof df the exactness of the sequence of SPANIER-WHITEHEAD in
S-theoryl.
Chapter VI is chiefly an application of the preceding chapter to
simplicial sets and to the homotopic category. It also gives the definition
of homotopy groups, and various technical developments concerning
minimal jibrations, whose purpose is to prove (i) that every fibration is
homotopicallyequivalent (modulo the base) to a locally trivial morphism,
and (ii) the J. H. C. Whitehead theorem for simplicial sets.
Finally Chapter VII is limited to bringing together the preceding
material in order to prove the theorems referred to in the beginning of
this introduction, and which constitute the justification of the work
itself.
All this, as has already been said, is only an introduction to algebraic
topology. To make this introduction at least more or less complete, we
have sketched briefly in two appendices a few complementary remarks
of interest to the reader.
In Appendix I will be found a theory of coverings and local systems,
and as an application a proof of the Van Kampen theorem for the geo
metric realization of simplicial sets.
In Appendix II the reader will find, as a bonus, a version of ElLEN
BERG'S theorem connecting the homology of a complex with the singular
1 The theory of carriers and S -theory - Algebraic geometry and Topo
logy. A symposium in honor of S. LEFSCHETZ. Princeton University Press,
1956, pp. 330-360.
VIII Introduction
homology of its geometrical realization, and the spectral sequence of a
fibration.
The present volume is an outgrowth of a seminar given by the authors
in 1963/64 in the Institut de Mathematique de Strasbourg, with the help
of C. GODBILLON.
We wish to thank Professor A. DOLD who asked us to write this book
for Springer-Verlag, and Professor P. HILTON who read the first draft
of this book and accepted it for publication in the Ergebnisse series.
We thank also Mr. Luc DEMERS qui a ete charge de la tache ingrate
de traduire Ie manuscript dans la langue d'outre-Manche.
Strasbourg, 1. 6. 1966 P. GABRIEL, M. ZISMAN
Contents
Introduction V
Leitfaden (Schema) X
Dictionary 1
Chapter I. Categories of Fractions 6
1. Categories of Fractions. Categories of Fractions and Adjoint
Functors. . . . . . . . . . . . 6
2. The Calculus of Fractions 11
3. Calculus of Left Fractions and Direct Limits 16
4. Return to Paragraph 1 19
Chapter II. Simplicial Sets 21
1. Functor Categories 21
2. Definition of Simplicial Sets 23
3. Skeleton of a Simplicial Set. 26
4. Simplicial Sets and Category of Categories 31
5. Ordered Sets and Simplicial Sets. Shuffles 33
6. Groupoids . . . . . . . . . . . . . . 35
7. Groupoids and Simplicial Sets . . . . . 38
Chapter III. Geometric Realization of Simplicial Sets 41
1. Geometric Realization of a Simplicial Set 41
4. Kelley Spaces. . . . . . . . . . . . . . 47
3. Exactness Properties of the Geometric Realization Functor. 49
4. Geometric Realization of a Locally Trivial Morphism. 54
Chapter IV. The Homotopic Category. 57
1. Homotopies 57
2. Anodyne Extensions 60
3. Kan Complexes . . 65
4. Pointed Complexes 69
5. Poincare Group of a Pointed Complex 76
Chapter V. Exact Sequences of Algebraic Topology 78
1. 2-Categories ... . . . . . . . . . 78
2. Exact Sequences of Pointed Groupoids. . . 82
3. Spaces of Loops. . . . . . . . . . . . . 84
4. Exact Sequences: Statement of the Theorem and Invariance 88
5. Proof of Theorem 4.2 . . . . . . . . . 91
6. Duality . . . . . . . . . . . . . . . . . . . . . .. 96
7. First Example: Pointed Topological Spaces. . . . . . .. 100
8. Second Example: Differential Complexes of an Abelian Category 102
Chapter VI. Exact Sequences of the Homotopic Category 106
1. Spaces of Loops. . 106
2. Cones . . . . . . . . . 111
3. Homotopy Groups. . . . 116
4. Generalities on Fibrations 119
5. Minimal Fibrations 124
x Contents
Chapter VII. Combinatorial Description of Topological Spaces 131
1. Geometric Realization of the Homotopic Category. . 131
2. Geometric Realization of the Pointed Homotopic Category 13S
3. Proof of MILNOR'S Theorem. 137
Appendix I. Coverings . . . . . . . . . . . . . . . 139
1. Coverings of a Groupoid . . . . . . . . . . . 139
2. Coverings of Groupoids and Simplicial Coverings 141
3. Simplicial Coverings and Topological Coverings . 144
Appendix II. The Homology Groups of a Simplicial Set. 148
1. A Theorem of ElLENBERG • • . • • . . . . . 148
2. The Reduced Homology Group of a Pointed Simplicial Set 1 S1
3. The Spectral Sequence of Direct Limits 1 S3
4. The Spectral Sequence of a Fibration 1 S7
Bibliography 163
Index of Notations . 165
Terminological Index 167
leiI fa'del7
Dictionary
The aim of this dictionary is to define with precision the terms which
will be used in the sequel. For the basic notions, we refer the reader to
the following works:
GABRIEL, P.: Categories abeliennes. Bull. Soc. Math. France 90
(1962). GROTHENDIECK, A.: Sur quelques points d'algebre homologique.
Tohoku math. J. serie 2. 9 (1957). MACLANE, S.: Homology. Berlin
Heidelberg-New York: Springer. MITCHELL, B.: Theory of Categories.
New York: Academic Press.
Unfortunately, the terminologies used in these books coincide neither
with each other, nor with those which we will sometimes use. It is this
great variety of language which forces us to restrict the number of
publications given as references.
Adjoint: See GABRIEL (op. cit.) for the notations. We say that T
is left adjoint to S and that S is right adjoint to T. We say that 1p is an
adjunction isomorphism from T to S, that lJI is an adjunction morphism
from T to S; similarly, rp is an adjunction isomorphism from S to T,
and fjj is an adjunction morphism from S to T.
We say that lJI is quasi inverse to fjj, and conversely.
Amalgamated sum: It is equivalent to the expression "somme
fibree" of GABRIEL (loc. cit.) and pushout of MITCHELL (loc. cit.). We
C a, b
write A 11 B or A 11 B for the amalgamated sum of a diagram of the
form
Arrow: See category and diagram scheme.
Can: Short for canonical.
Category: See the references. If rc is a category, we will write Db rc
(resp. 'ltt rc) for the class of its objects (resp. morphisms or arrows). The
identity morphism of an object c of a category rc will be denoted by Id'6'c,
or simply by Id c. If I is a morphism of a category C, the domain and the
range of I will be denoted by '0'6'1 and t'6'l, or simply by '01 and t/. The set
of morphisms of a category rc with domain a and range b will be denoted
by rc(a, b), or Hom'6' (a, b).
Category of paths of a diagram scheme T: It is a category f!JJa T whose
objects are the same as those of T, and whose morphisms are the se-
