Acyclic Models

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Brand new. We distribute directly for the publisher. Acyclic models is a method heavily used to analyze and compare various homology and cohomology theories appearing in topology ... and algebra. This book is the first attempt to put together in a concise form this important technique and to include all the necessary background.It presents a brief introduction to category theory and homological algebra. The author then gives the background of the theory of differential modules and chain complexes over an abelian category to state the main acyclic models theorem, generalizing and systemizing the earlier material. This is then applied to various cohomology theories in algebra and topology.The volume could be used as a text for a course that combines homological algebra and algebraic topology. Required background includes a standard course in abstract algebra and some knowledge of topology. The volume contains many exercises. It is also suitable as a reference work for researchers.Titles in this series are co-published with the Centre de Recherches Mathmatiques. Read more Show Less

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Overview

Acyclic models is a method heavily used to analyze and compare various homology and cohomology theories appearing in topology and algebra. This book is the first attempt to put together in a concise form this important technique and to include all the necessary background. It presents a brief introduction to category theory and homological algebra. The author then gives the background of the theory of differential modules and chain complexes over an abelian category to state the main acyclic models theorem, generalizing and systemizing the earlier material. This is then applied to various cohomology theories in algebra and topology. The volume could be used as a text for a course that combines homological algebra and algebraic topology. Required background includes a standard course in abstract algebra and some knowledge of topology. The volume contains many exercises. It is also suitable as a reference work for researchers.

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Editorial Reviews

From The Critics
Provides an introductory discussion of category theory; abelian categories and homological algebra; the homology of chain complexes and discussion of simplicial objects and the associated chain complexes; the part of the theory of triples that is needed for acyclic models; proof for the main acyclic models theorem with conclusions; and three chapters providing applications of the theory. Requires familiarity with basic algebra and topology. Suitable for use as a text for a course combining homological algebra and algebraic topology. Includes exercises. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780821828779
  • Publisher: American Mathematical Society
  • Publication date: 6/1/2002
  • Series: CRM Monograph Series , #17
  • Pages: 179
  • Product dimensions: 7.20 (w) x 10.20 (h) x 0.60 (d)

Table of Contents

Preface
Ch. 1 Categories 1
1 Introduction 1
2 Definition of category 1
3 Functors 9
4 Natural transformations 12
5 Elements and subobjects 15
6 The Yoneda Lemma 19
7 Pullbacks 22
8 Limits and colimits 26
9 Adjoint functors 35
10 Categories of fractions 38
11 The category of modules 42
Ch. 2 Abelian Categories and Homological Algebra 45
1 Additive categories 45
2 Abelian categories 49
3 Exactness 51
4 Homology 56
5 Module categories 61
6 The Z construction 67
Ch. 3 Chain Complexes and Simplicial Objects 69
1 Mapping cones 69
2 Contractible complexes 72
3 Simplicial objects 76
4 Associated chain complex 80
5 The Dold-Puppe theorem 81
6 Double complexes 82
7 Double simplicial objects 86
8 Homology and cohomology of a morphism 87
Ch. 4 Triples a la Mode de Kan 89
1 Triples and cotriples 89
2 Model induced triples 92
3 Triples on the simplicial category 93
4 Historical Notes 94
Ch. 5 The Main Acyclic Models Theorem 95
1 Acyclic classes 95
2 Properties of acyclic classes 99
3 The main theorem 101
4 Homotopy calculuses of fractions 104
5 Exactness conditions 109
Ch. 6 Cartan-Eilenberg Cohomology 113
1 Beck modules 114
2 The main theorem 119
3 Groups 122
4 Associative algebras 125
5 Lie Algebras 126
Ch. 7 Other Applications in Algebra 131
1 Commutative algebras 131
2 More on cohomology of commutative cohomology 145
3 Shukla cohomology 148
4 The Eilenberg-Zilber theorem 148
Ch. 8 Applications in Topology 151
1 Singular homology 151
2 Covered spaces 156
3 Simplicial homology 160
4 Singular homology of triangulated spaces 161
5 Homology with ordered simplexes 163
6 Application to homology on manifolds 168
Bibliography 175
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