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We define a localization L of a category E to be quintessential if the left adjoint to the inclusion functor is also right adjoint to it, and persistent if L is closed under subobjects in E. We show that quintessential localizations of an arbitrary Cauchy-complete category correspond to idempotent natural endomorphisms of its identity functor, and that they are necessarily persistent. Our investigation of persistent localizations is largely restricted to the case when E is a topos: we show that persistence is equivalence to the closure of L under finite coproducts and quotients, and that it implies that L is coreflective as well as reflective, at least provided E admits a geometric morphism to a Boolean topos. However, we provide examples to show that the reflector and coreflector need not coincide.
@article{TAC_1996_2_a7,
author = {P.T. Johnstone},
title = {Remarks on {Quintessential} and {Persistent} {Localizations}},
journal = {Theory and applications of categories},
pages = {90--99},
publisher = {mathdoc},
volume = {2},
year = {1996},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_1996_2_a7/}
}
P.T. Johnstone. Remarks on Quintessential and Persistent Localizations. Theory and applications of categories, Tome 2 (1996), pp. 90-99. http://geodesic.mathdoc.fr/item/TAC_1996_2_a7/