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Some Remarks on Limits in Categories

Published online by Cambridge University Press:  20 November 2018

J. M. Maranda
J. M. Maranda*
Affiliation:
Université de Montréal
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The object of this paper is to give simple criteria for the existence of direct limits in categories and for the permuting of a functor with direct limits.

The notion of direct limit of a diagram that we shall use here is essentially that of Kan (4), which is more general than the usual notion of direct limit of a directed diagram.

Our treatment is based on the fact (Lemma 2) that the usual process for constructing the direct limit of a diagram of modules, which consists in taking a direct sum of the modules in the diagram and then considering a certain homomorphic image of this direct sum (3, p. 220), is essentially, once certain notions have been properly generalized, the only process for constructing the direct limit of any diagram in any category.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 5 , Issue 2 , May 1962 , pp. 133 - 146
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Bourbaki, N., Théorie des ensembles, Chap. 4, Act. Sci. et Ind., no. 1258, Paris 1957.Google Scholar
2. Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, 1956.Google Scholar
3. Eilenberg, S. and Steenrod, N., Foundations of Algebraic Topology, Princeton University Press, 1952.10.1515/9781400877492Google Scholar
4. Kan, D. M., Adjoint Functors, Trans, of the Am. Math. Soc., Vol. 87 (1958), pp. 294-329.Google Scholar