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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.
8
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Accepted
The Mittag-Leffler condition as necessary and sufficient
This is due to Emmanouil as far as I know. See this.