Accessibility and the solution set condition Original Research Article

It is shown that accessibility of functors between accessible categories is fully characterized by Freydʹs solution set condition, provided the set-theoretic Vopimagenka principle holds. The Yoneda embedding of accessible categories satisfies the solution set condition; under Vopimagenkaʹs principle, this is true for all categories with a small dense generator. In each case, Vopimagenkaʹs principle is in fact equivalent to the validity of the desired results.