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On the ordered set of reflective subcategories

Published online by Cambridge University Press:  17 April 2009

G. M. Kelly
G. M. Kelly
Affiliation:
Pure Mathematics Department, University of Sydney, New South Wales, 2006, Australia.
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Abstract

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Given a category A, we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is complete and cocomplete, wellpowered and cowellpowered, the intersection of two reflective subcategories need not be reflective. Supposing that A admits (i) small limits and (ii) arbitrary (even large) intersections of strong subobjects, we prove that an infimum ∧iCi in Ref A must necessarily be the intersection ∩iCi. Accordingly Ref A is not in general, even for good A, a complete lattice. We show, however, under the same conditions on A, that Ref A does admit small suprema ∨iCi, given by the closure in A of the union ∪iCi under the limits of type (i) and (ii) above.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 1 , August 1987 , pp. 137 - 152
Copyright
Copyright © Australian Mathematical Society 1987

References

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