Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T18:18:54.873Z Has data issue: false hasContentIssue false
  • English
  • Français

A Relationship between Left Exact and Representable Functors

Published online by Cambridge University Press:  20 November 2018

H. B. Stauffer
H. B. Stauffer*
Affiliation:
University of Chicago, Chicago, Illinois University of British Columbia, Vancouver, British Columbia California State College, Hayward, California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our aim in this paper is to demonstrate a relationship between left exact and representable functors. More precisely, in the functor category whose objects are the additive functors from the dual of an abelian category 𝔄 to the category of abelian groups and whose morphisms are the natural transformations between them, the left exact functors can be characterized as those equivalent to a direct limit of representable functors taken over a directed class. The proof will proceed in the following manner. Lambek [3] and Ulmer [7] have shown that any functor T in can be expressed as a direct limit of representable functors taken over a comma category. When T is left exact, it is easily observed that this comma category is a filtered category. When T is left exact, it is easily observed that this comma category is a filtered category.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 374 - 380
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Freyd, P., Abelian categories (Harper and Row, New York, 1964).Google Scholar
2. Grothendieck, A., Technique de descente et théorèmes d'existence en géométrie algébrique. Il: Le théorème d'existence en théorie formelle des modules, Séminaire Bourbaki, 1959/1960, Tome 12, Fasc. 2, Exposé 195, 22 pp. (Secrétariat mathématique, Paris, 1960).Google Scholar
3. Lambek, J., Completions of categories (Springer, Berlin, 1966).Google Scholar
4. Lawvere, F. W., The category of categories as a foundation for mathematics, Proc. Conf. Categorical Algebra, La Jolla, California, 1965, pp. 1–20 (Springer, New York, 1966).Google Scholar
5. Mitchell, B., Theory of categories (Academic Press, New York, 1965).Google Scholar
6. Ulmer, F., Satelliten und derivierte funktoren. I, Math. Z. 91 (1966), 216–266.Google Scholar
7. Ulmer, F., Properties of dense and relative adjoint functors, J. Algebra 8 (1968), 77–95.Google Scholar