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Acyclic Models

Published online by Cambridge University Press:  20 November 2018

Michael Barr
Michael Barr*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, email: barr@math.mcgill.ca
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Abstract

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Acyclic models is a powerful technique in algebraic topology and homological algebra in which facts about homology theories are verified by first verifying them on "models" (on which the homology theory is trivial) and then showing that there are enough models to present arbitrary objects. One version of the theorem allows one to conclude that two chain complex functors are naturally homotopic and another that two such functors are object-wise homologous. Neither is entirely satisfactory. The purpose of this paper is to provide a uniform account of these two, fixing what is unsatisfactory and also finding intermediate forms of the theorem.

Keywords

18G3555U1555N40

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 2 , 01 April 1996 , pp. 258 - 273
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. André, M.,Méthode Simpliciale en Algébre Homologique et Algébre Commutative, Lecture Notes in Math. 32, Springer-Verlag, Berlin, Heidelberg, New York, 1967.Google Scholar
2. Appelgate, H.,Acyclic models and resolvent functors, Dissertation, Columbia University, 1965.Google Scholar
3. Ban*, M., Cartan-Eilenberg cohomology and triples, J. Pure Appl. Algebra, to appear.Google Scholar
4. Barr, M. and Beck, J., Acyclic models and triples, (eds. Eilenberg, S. et. al.), Proc. Conf. Categorical Algebra, La Jolla, 1965. Springer-Verlag, 1966. 336–343.Google Scholar
5. Dold, A.,Lectures on Algebraic Topology, Springer-Verlag, Berlin, Heidelberg, New York, 1980.Google Scholar
6. Eilenberg, S., Singular homology theory, Ann. Math. 45 (1944), 207—247.Google Scholar
7. Eilenberg, S. and MacLane, S., General theory of natural equivalences, Trans. Amer. Math. Soc. 58(1945), 231–244.Google Scholar
8. Eilenberg, S., Acyclic Models, Amer. J. Math. 75 (1953), 189–199.Google Scholar
9. Kleisli, H., On the contraction of standard complexes, J. Pure Appl. Algebra 4 (1974), 243–260.Google Scholar
10. Lefschetz, S., Algebraic Topology, Amer. Math. Soc. Colloq. Publ. 27, 1942.Google Scholar