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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.
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Tips on cohomology for number theory
I'd seriously recommend Mazur's "notes on etale cohomology of number fields" survey for a nice balance between formal manipulations and geometric intuition. Also at a more basic level, try Neukrich's …