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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.
3
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Tensor product of mapping cones
Th answer by White is not quite right, even for short exact sequences of vector spaces (for dimension reasons).
One way to describe $C_1 \otimes^L C_2$ is via filtrations. Regard each of the given ex …