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A notion of limit for enriched categories

Published online by Cambridge University Press:  17 April 2009

Francis Borceux  and
G.M. Kelly
Francis Borceux
Affiliation:
Départment de Mathématiques, Université de Louvain, Louvain-la-Neuve, Belgium.
G.M. Kelly
Affiliation:
Department of Pure Mathematiques, University of Sydney, Sydney, New South Wales.
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Abstract

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For a V-category B, where V is a symmetric monoidal closed category, various limit-like notions have been recognized: ordinary limits (in the underlying category B0 ) preserved by the V-valued representable functors; cotensor products; ends; pointwise Kan extensions. It has further been recognized that, to be called complete, B should admit all of these; for which it suffices to demand the first two. Hitherto, however, there has been no single limit-notion of which all these are special cases, and particular instances of which may exist even when B is not complete or even cotensored. In consequence it has not been possible even to state, say, the representability criterion for a V-functor T: B → V, or even to define, say, pointwise Kan extensions into B, except for cotensored B. (It is somewhat as if, for ordinary categories, we had the notions of product and equalizer, but lacked that of general limit, and could not discuss pullbacks in the absence of products.) In this paper we provide such a general limit-notion for V-categories.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1975 , pp. 49 - 72
Copyright
Copyright © Australian Mathematical Society 1975

References

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