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nPOV: Cohomology and derived functors
In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. Namely,
$$H^n(X,Y) = \pi_0 \mathbb{H}(X,B^nY),$$
where $\mathbb{H}$ is an $(\infty,1)$-topos and $B$ is the …
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Unifying "cohomology groups classify extensions" theorems
This is not an answer, but just some pointers to things I guess are related! I posted this in the comments, but it got really messy. So I moved it here.
-- original comments below --
I also want to …