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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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Splitness of commutative diagrams
I don't know if this will help, but this is what I make of it. First, clearly necessary conditions will be that top and bottom rows split. Thus we may as well suppose that the top row is $$0\to C\ma …