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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.
4
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Is the tensor product of chain complexes a Day convolution?
Note: I'm not sure the construction I give is correct. There is an unresolved problem in proving it is a comonoid. See point 4 in the comonoid section. I don't have the time to resolve this right now. …