**CRM Monograph Series**

Volume: 17;
2002;
179 pp;
Hardcover

MSC: Primary 18;

Print ISBN: 978-0-8218-2877-9

Product Code: CRMM/17

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

**Electronic ISBN: 978-1-4704-3862-3
Product Code: CRMM/17.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

# Acyclic Models

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*Michael Barr*

A co-publication of the AMS and Centre de Recherches Mathématiques

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 Mathématiques.

#### Readership

Graduate students and research mathematicians interested in category theory and homological algebra.

#### Reviews & Endorsements

I like this book. It covers ground not often explored in textbooks … with a perspective … complementary to the usual.

-- Zentralblatt MATH

This much needed book provides a clear and self-contained account of the cotriple approach to the cohomology theory of algebraic objects. Written by one of the founders of the subject, it will prove useful both as a teaching text and reference text for researchers.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Acyclic Models

- Cover Cover11
- Title page i2
- Dedication iii4
- Contents v6
- Preface vii8
- Categories 114
- Abelian categories and homological algebra 4558
- Chain complexes and simplicial objects 6982
- Triples à la mode de Kan 89102
- The main acyclic models theorem 95108
- Cartan-Eilenberg Cohomology 113126
- Other applications in algebra 131144
- Applications in topology 151164
- Bibliography 175188
- Index 177190
- Back Cover Back Cover1194