[002G] Theorem 5·f (Pro-localization conditions).
  • For every functor we have the implications

FF is very rich \To FF is rich (Hilton) \To FF is opaque \To FF is consistent (Frei)

and each converse implication is false in general.

  • An opaque functor is consistent and co-consistent (i.e. FopF^\op is consistent).
  • If LUL\dashv U is an adjunction, UU is opaque if and only if LL is opaque.
  • If LUL\dashv U is an adjunction, UU is very rich if and only if it is rich, if and only if it is opaque, if and only if it is consistent.