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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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1-1 map on the $\{0,1\}^k$
Let integer $k>0$ and let $\{0,1\}^k$ denote the set of all $1\times k$-dim vectors whose every coordinate is eithor 0 or 1, for example, $(0,1,1,0,\dots,1,0,0,1)$. For any such vector $\alpha$, let $ …