Acyclic Models

Acyclic Models

by Michael Barr
     
 

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

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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.

Product Details

ISBN-13:
9780821828779
Publisher:
American Mathematical Society
Publication date:
06/01/2002
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. 1Categories1
1Introduction1
2Definition of category1
3Functors9
4Natural transformations12
5Elements and subobjects15
6The Yoneda Lemma19
7Pullbacks22
8Limits and colimits26
9Adjoint functors35
10Categories of fractions38
11The category of modules42
Ch. 2Abelian Categories and Homological Algebra45
1Additive categories45
2Abelian categories49
3Exactness51
4Homology56
5Module categories61
6The Z construction67
Ch. 3Chain Complexes and Simplicial Objects69
1Mapping cones69
2Contractible complexes72
3Simplicial objects76
4Associated chain complex80
5The Dold-Puppe theorem81
6Double complexes82
7Double simplicial objects86
8Homology and cohomology of a morphism87
Ch. 4Triples a la Mode de Kan89
1Triples and cotriples89
2Model induced triples92
3Triples on the simplicial category93
4Historical Notes94
Ch. 5The Main Acyclic Models Theorem95
1Acyclic classes95
2Properties of acyclic classes99
3The main theorem101
4Homotopy calculuses of fractions104
5Exactness conditions109
Ch. 6Cartan-Eilenberg Cohomology113
1Beck modules114
2The main theorem119
3Groups122
4Associative algebras125
5Lie Algebras126
Ch. 7Other Applications in Algebra131
1Commutative algebras131
2More on cohomology of commutative cohomology145
3Shukla cohomology148
4The Eilenberg-Zilber theorem148
Ch. 8Applications in Topology151
1Singular homology151
2Covered spaces156
3Simplicial homology160
4Singular homology of triangulated spaces161
5Homology with ordered simplexes163
6Application to homology on manifolds168
Bibliography175

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