The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory, i.e. its inclusion functor has a left-exact left adjoint. Can this left adjoint ever have a further left adjoint, so that the hypercomplete objects would be an essential localization or "level"?
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Jacob's answer to my last question actually answers this one too: if the hypercomplete objects are coreflective as well as reflective, and the coreflector is the same functor as the reflector, then of course the reflector also has a left adjoint, namely the inclusion functor.
Mike Shulman
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