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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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"Sums-compact" objects = f.g. objects in categories of modules?
Group of rational numbers is sumpact. In older terminology sumpact for abelian categories is called small.