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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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"Pick up a homological algebra book and prove all of the theorems yourself" (exercise from L...
I got to this list since I was trying to remember whether the quote from Lang was true or not. Indeed, my copy, first edition I guess, does have that quote on p. 105. I don't know exactly how "posit …