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If you have two triangulated functors $F,G$, you can compute the cohomology of $FG(-)$ in terms of the cohomology of $G(-)$ and the cohomology of $F(-)$, in terms of a ``Grothendieck spectral sequence''. e.g. if $F,G$ are right derived functors on derived categories. This just comes from the standard spectral sequence on filtered double complexes.

Question: Is there anything written about ``$\ge 3$ dimensional'' spectral sequences?

By this I mean something which computes the cohomology of e.g. $FGH(-)$ in terms of cohomologies of the constituent functors, or computes the cohomology of a triple complex with (two?) filtrations. I'm aware that in principle you could just apply the spectral sequence to $F,G$ then to $FG,H$, but I imagine that it might be cleaner to do it all in one go and it would be nice to have a reference.

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    $\begingroup$ Here's a possibly even more mindblowing concept: what sort of thing is the difference between the spectral sequences gotten from the comparisons in the second compositions in each of $(FG)H$ and $F(GH)$? $\endgroup$
    – David Roberts ♦
    Commented Jun 1, 2021 at 11:21
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    $\begingroup$ See arxiv.org/pdf/1308.3187v1.pdf $\endgroup$
    – Drew Heard
    Commented Jun 1, 2021 at 11:24
  • $\begingroup$ @Meow : do you have a reference the for Grothendieck spectral sequence that applies to complexes (and not only objects) of an abelian category ? I am interested. $\endgroup$
    – Niels
    Commented Jun 1, 2021 at 12:58
  • $\begingroup$ @Niels Sadly not, the link (mathoverflow.net/questions/24143/…) I posted only works when the object $X$ is pure. $\endgroup$
    – Pulcinella
    Commented Jun 1, 2021 at 13:06
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    $\begingroup$ @Meow section 2 of this is relevant: link.springer.com/article/10.1007/s40062-015-0120-0 $\endgroup$
    – Ishan Levy
    Commented Jun 14, 2021 at 19:43

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