  We define and study sl\_2-categorifications on abelian categories. We show in
particular that there is a self-derived (even homotopy) equivalence
categorifying the adjoint action of the simple reflection. We construct
categorifications for blocks of symmetric groups and deduce that two blocks are
splendidly Rickard equivalent whenever they have isomorphic defect groups and
we show that this implies Brou\'e's abelian defect group conjecture for
symmetric groups. We give similar results for general linear groups over finite
fields. The constructions extend to cyclotomic Hecke algebras. We also
construct categorifications for category O of gl\_n(C) and for rational
representations of general linear groups over an algebraically closed field of
characteristic p, where we deduce that two blocks corresponding to weights with
the same stabilizer under the dot action of the affine Weyl group have
equivalent derived (and homotopy) categories, as conjectured by Rickard.
