  We define highest weight categorical actions of sl_2 on highest weight
categories and show that basically all known examples of categorical
sl_2-actions on highest weight categories (including rational and polynomial
representations of general linear groups, parabolic categories O of type $A$,
categories O for cyclotomic Rational Cherednik algebras) are highest weight in
our sense. Our main result is an explicit combinatorial description of (the
labels of) the crystal on the set of simple objects. A new application of this
is to determining the supports of simple modules over the cyclotomic Rational
Cherednik algebras starting from their labels.
