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.


1991 MSC: 18A40, 18B25.

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

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