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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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What is a cohomology theory (seriously)?
One can give a unified meaning to "cohomology theories" on any ground category by making use of a canonical axiomatic setting: families of functors with values in suitable target categories satisfying …