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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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Simplicial Complex Induced by a Morphism
Have you looked at Illusie's Complexe Cotangent et Deformations I? Specifically Section 1.3, "The Theorem of Dold-Puppe" where there's a fairly formal formula for the simplicial object to which your c …