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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 22nd 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexFrom [[supercompact cardinal]]: ---- **Theorem:** The existence of arbitrarily large supercompact cardinals implies the statement: Every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if $L$ is a [[reflective subcategory|reflective]] [[localization]] functor on a [[balanced category|balanced]] [[accessible category]] such that the unit morphism $X \to L X$ is an [[epimorphism]] for all $X$ and the [[class]] of $L$-[[local object]]s is defined by an [[absolute formula]], then the existence of a suficciently large supercompact cardinal implies that $L$ is a localization with respect to some [[set]] of morphisms. This is in _BagariaCasacubertaMathias_ > [[Urs Schreiber]]: I am being told in prvivate communication that the assumption of epis can actually be dropped. A refined result is due out soon. ----- does anyone know about this refined result?

   From supercompact cardinal:

   ---

   Theorem: The existence of arbitrarily large supercompact cardinals implies the statement:

   Every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class.

   In other words, if $L$ is a reflective localization functor on a balanced accessible category such that the unit morphism $X \to L X$ is an epimorphism for all $X$ and the class of $L$-local objects is defined by an absolute formula, then the existence of a suficciently large supercompact cardinal implies that $L$ is a localization with respect to some set of morphisms.

   This is in BagariaCasacubertaMathias

   Urs Schreiber: I am being told in prvivate communication that the assumption of epis can actually be dropped. A refined result is due out soon.

   ---

   does anyone know about this refined result?