categorification
Many important cases of categorification (in fact most of those so far studied in $n$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 $sl_2$-categorification, Ann. Math. 167 (2008), 245–298.
S.-J. Kang, M. Kashiwara, Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras, arXiv:1102.4677
S.-J. Kang, M. Kashiwara, E. Park, Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data, arxiv/1202.1622
M. Kashiwara, Biadjointness in cyclic Khovanov-Lauda-Rouquier algebras, arxiv/1111.5898
A. Lauda, M. Vazirani, Crystals from categorified quantum groups, Adv. Math., 228, no. 2, (2011), 803–861, arXiv:0909.1810
Mikhail Khovanov, Nilcoxeter algebras categorify the Weyl algebra, Comm. Algebra 29, No. 11 (2001) 5033–5052, math.RT/9906166, MR2002h:16041, doi
Mikhail Khovanov, V. Mazorchuk, Catharina Stroppel, A brief review of abelian categorifications, math.RT/0702746
J. Bernstein, I. Frenkel, M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(sl_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), 199-241, MR2000i:17009, doi
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Mathematical Journal 101 (3): 359–426, 2000, doi, MR1740682
Igor Frenkel, Catharina Stroppel, Joshua Sussan, Categorifying fractional Euler characteristics, Jones-Wenzl projector and $3j$-symbols, arXiv:1007.4680
Catharina Stroppel, Joshua Sussan, Categorified Jones-Wenzl Projectors: a comparison, arXiv:1105.3038
Catharina Stroppel, Ben Webster, Quiver Schur algebras and q-Fock space, arXiv:1110.1115
Ivan Losev, Ben Webster, On uniqueness of tensor products of irreducible categorifications, arxiv/1303.1336