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.
Author(s): Michael Barr
Series: CRM Monograph Series 17
Publisher: American Mathematical Society, Centre de Recherches Mathématiques
Year: 2017
Language: English
Pages: 199
Preface......Page 7
2. Definition of category......Page 13
3. Functors......Page 21
4. Natural transformations......Page 25
5. Elements and subobjects......Page 28
6. The Yoneda Lemma......Page 32
7. Pullbacks......Page 35
8. Limits and colimits......Page 39
9. Adjoint functors......Page 48
10. Categories of fractions......Page 51
11. The category of modules......Page 56
12. Filtered colimits......Page 57
1. Additive categories......Page 60
2. Abelian categories......Page 64
3. Exactness......Page 66
4. Homology......Page 72
5. Module categories......Page 76
6. The construction......Page 83
1. Mapping cones......Page 84
2. Contractible complexes......Page 87
3. Simplicial objects......Page 91
4. Associated chain complex......Page 95
5. The Dold-Puppe theorem......Page 97
6. Double complexes......Page 98
7. Double simplicial objects......Page 102
8. Homology and cohomology of a morphism......Page 103
1. Triples and cotriples......Page 105
2. Model induced triples......Page 108
3. Triples on the simplicial category......Page 109
4. Historical Notes......Page 110
1. Acyclic classes......Page 112
2. Properties of acyclic classes......Page 116
3. The main theorem......Page 118
4. Homotopy calculuses of fractions......Page 121
5. Exactness conditions......Page 126
Chapter 6. Cartan–Eilenberg cohomology......Page 129
1. Beck modules......Page 130
2. The main theorem......Page 136
3. Groups......Page 138
4. Associative algebras......Page 141
5. Lie Algebras......Page 142
1. Commutative Algebras......Page 148
2. More on cohomology of commutative cohomology......Page 163
4. The Eilenberg–Zilber theorem......Page 166
1. Singular homology......Page 169
2. Covered spaces......Page 174
3. Simplicial homology......Page 178
4. Singular homology of triangulated spaces......Page 180
5. Homology with ordered simplexes......Page 181
6. Application to homology on manifolds......Page 187
Bibliography......Page 194
Index......Page 197
