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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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Derived functors and functorial resolutions/(co)fibrant replacements
I will begin with some context; the question itself is highlighted below. This is all for some notes I am writing personally on homological algebra, amongst other things.
To construct derived functors …