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All sorts of things are categorified. What about spectral sequences?

Question: What is a categorification of a spectral sequence?

Talking through my hat, I could imagine an $\infty$-category (categorification of the abelian category of spectral sequences) with an $\infty$-functor (limit) into another $\infty$-category (categorification of homotopy or cohomology). Any pointers will be greatly appreciated.

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    $\begingroup$ If one takes the view that the data of a spectral sequence is equivalent to the data of the filtered object, the following might be relevant. arxiv.org/abs/2109.01017 $\endgroup$
    – Chris H
    Commented Jun 27 at 8:08

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