#
Remarks on Quintessential and Persistent Localizations

##
P.T. Johnstone

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.

Keywords:

1991 MSC: 18A40, 18B25.

*Theory and Applications of Categories*, Vol. 2, 1996, No. 8, pp 90-99.

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