Remarks on Quintessential and Persistent Localizations

We de ne 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 i dentity 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 nite coproducts and quotients and that it implies that L is core ective as well as re ective at least provided E admits a geometric morphism to a Boolean topos However we provide examples to show that the re ector and core ector need not coincide