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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.

2 votes

Confusion about signs in the definition of an $A_\infty$-algebra

Alexander Polishchuk has written a Field Guide to $A_\infty$ Sign Conventions which I have found quite helpful (despite some typos and imprecisions). Interestingly, Raf Bocklandt gives this as the ma …
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