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(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
9
votes
Beilinson-Bernstein and Koszul duality
I was first introduced to Soergel's conjecture as a "categorification" of Vogan duality which identifies blocks of representations for a real reductive Lie group (a few more adjectives should be inclu …