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Acyclic Models
Michael Barr, Peter Redpath Emeritus Professor of Pure Mathematics, McGill University, Montreal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Monograph Series
2002; 179 pp; hardcover
Volume: 17
ISBN-10: 0-8218-2877-0
ISBN-13: 978-0-8218-2877-9
List Price: US$60
Member Price: US$48
Order Code: CRMM/17
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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.


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


"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

  • Categories
  • Abelian categories and homological algebra
  • Chain complexes and simplicial objects
  • Triples à la mode de Kan
  • The main acyclic models theorem
  • Cartan-Eilenberg Cohomology
  • Other applications in algebra
  • Applications in topology
  • Bibliography
  • Index
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