The geometric Langlands program
:* geometric Satake?
Many important cases of categorification (in fact most of those so far studied in Lab) belong to the categorification of basic and general structures in category theory, algebra and geometry like fibered categories, monads, operads, sheaves etc. To find the “correct” categorification one usually needs just clear understanding of foundations and clear categorical strategy.
On the other hand, a number of categorifications of rather special structures in representation theory on the interface of Lie theory and low dimensional topology, is emerging from study of rather special and deep phenomena. In those examples special and often advanced structures in quantum group theory, knot theory etc. are starting revealing to be a shadow of more fundamental structures on the categorified level.
J. Chuang, R. Rouquier, Derived equivalences for symmetric groups and -categorification, Ann. Math. 167 (2008), 245–298.
J. Bernstein, I. Frenkel, M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), 199-241, MR2000i:17009, doi
Ivan Losev, Ben Webster, On uniqueness of tensor products of irreducible categorifications, arxiv/1303.1336