I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange).

I have been "brought up" as an algebraic geometer. Spectral sequences are obviously ubiqutous and useful in this subject. The conclusion I have drawn from my exposure to spectral sequences there is that they express how you can compute ordinary (non-derived) invariants of "derived objects". An alternative way of saying this that whenever you have spectral sequence you should identify it as being either a grothendieck spectral sequence (you are really doing a derived a composition), or a hyper(co)homoloy spectral sequence (you legitimately want to know ordinary invariants of derived objects).

I have now begun studying (classical) stable homotopy theory, and there seems to be a bewildering set of spectral sequences. I cannot explain any of them in the above terms, but many of them "feel like they are close to being of the above type".

Let me give some examples. Consider the spectral sequence of a homotopy limit:

$\lim^* \pi_* E_\bullet \Rightarrow \pi_* \operatorname{holim} E_\bullet$

(I'm writing $*$ for all indices to avoid getting into details.) If you pretend that there is a nice functor $D\pi: SH \to DAb$ taking a spectrum to a chain complex with homology groups the homotopy groups of the spectrum ($h_* D\pi = \pi_*$) and which commutes with homotopy limits, then the above "is just the hyperhomology spectral sequence". Unfortunately I'm fairly sure $D\pi$ cannot exist.

If we keep up the pretense for a bit, we could try to say $Map(E, F) = RHom(D\pi E, D\pi F)$ (this is getting real silly now, since $D\pi$ is fully faithful and essentially surjective), and then the Atiyah-Hirzebruch spectral sequence also becomes "just a hyperhomology spectral sequence".

It seems similarly imaginable that the Atiyah-Hirzebruch spectral sequence is an incarnation of the Leray-Serre spectral sequence (for the inclusion $i: * \to X$), although I am less sure how to even put this in symbols.

I could go on; by dreaming up $D\pi$ (or a related gadget) to have various (eventually contradictory) properties many spectral sequences can be "interpreted" in this way. But enough woffling.

**Now to my real question.**

Is there a way in which sense can be made of these ideas? For example by replacing $DAb$ by a more complicated abelian category? Alternatively, is there a better organising principle for spectral sequences in algebraic topology?

*Notes*

If X is a topological abelian group (spectrum), then $D\pi X = N_\bullet Sing(X)$ (normalised chain complex of the singular simplicial abelian group of X) has some of the properties dreamed up above. Since the spectral sequences apply to topological abelian groups and their maps and in this case *reduce* to the hyperhomology spectral sequences I have shown, this perhaps explains why the topologists' spectral sequences feel familiar.

Thanks, Tom

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