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On Commutative Squares

Published online by Cambridge University Press:  20 November 2018

Johann B. Leicht
Johann B. Leicht*
Affiliation:
University of Toronto
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The following elementary facts about certain commutative diagrams, called "squares," are stated and proved in terms of abelian groups and their homomorphisms. However, they are valid for arbitrary abelian categories and can be proved also for them. This does not need to be shown, since every abelian category can be embedded into the category of abelian groups with preservation of exact sequences according to a result due to S. Lubkin (1). Proofs are often omitted or given only for one half of a theorem, the other half being dual to the first. Generalizations to a larger class of diagrams containing all finite commutative diagrams are possible.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 59 - 79
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Lubkin, Saul, Imbedding of abelian categories, Trans. Amer. Math. Soc, 97 (1960), 410–417.Google Scholar
2. Watts, Charles E., Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc, 11 (1960), 5–8.Google Scholar
3. MacLane, Saunders, Homology (to be published).Google Scholar
4. Hilton, P. J. and Ledermann, W., Homology and ringoids III, Proc. Camb. Phil. Soc, 56 (1960), 1–12.Google Scholar