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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.
-1
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Is the $\infty$-category $N_{dg}(\mathrm{Ch}(\mathcal{A}))$ presentable?
If $\mathcal{A}$ is a small Grothendieck abelian category, one could possibly argue this way: recall that presentable means accessible and closed under small colimits.
Since $\mathcal{A}$ is small, th …
2
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Extensions in a full subcategory
Let $\mathcal{C}$ be an abelian category (feel free to put more adjectives here) and $\mathcal{D}$ a full abelian subcategory closed under kernels and cokernels.
Then by definition for $A,B\in \mathca …