TITLE: origin of spectral sequences in algebraic topology
QUESTION [29 upvotes]: 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

REPLY [2 votes]: Here is how I look at the situation after thinking hard about the answers. I will say things in the language of triangulated categories and homotopy limits being somewhat imprecise. I suppose some hares would say that this is really about $(\infty,1)$-categories; unfortunately I don't know enough about these gadgets.
We are in the following situation: we have triangulated categories $D_1$ and $D_2$ together with a triangle functor $\alpha: D_1 \to D_2$ (additive, preserving distinguished triangles, etc). We would like to "explain" or "describe" the effect of $\alpha$ on objets. Clearly this necessitates first a way of describing objects in the first place.
The most conventional way I can think of is to use homological functors $\pi_i: D_i \to A_i$ into abelian categories (i.e. functors turning distinguished triangles into long exact sequences). I will write $\pi_{i*}X$ for the graded object $\pi_i(\Sigma^*X),$ where $\Sigma$ is the shift functor. Ideally, we would like a way to compute $\pi_{2*} \alpha X$ in terms of $\pi_{1*}X$ and perhaps some extra data. It is sort of clear that this cannot work in full generality, because $\pi_1$ can throw away arbitrary amounts of information.
So we need a more "internal" way of describing objects. One way to do this is using filtrations. That is, we associate to $X$ a sequence $\dots \to X_{k-1} \to X_k \to X_{k+1} \to \dots$ such that $X = \operatorname{holim} X_\bullet$ or $X = \operatorname{hocolim} X_\bullet.$ For this to be useful we need the "subquotients" $\operatorname{cone}(X_k \to X_{k+1})$ to be "nice", e.g. live in a subcategory we understand well.
The "internal" and "external" approach are not unrelated; for example a t-structure on $D_1$ has as heart an abelian subcategory, every object acquires a filtration and a "cofiltration" (I think), and the "subquotients" live in the abelian subcategory, effectively giving us a compatible $\pi_1.$
So the strategy to describe $\alpha(X)$ is now this:

Find the (co)filtration $X_\bullet$ and relate $\alpha(\operatorname{ho(co)lim}X_\bullet)$ to $\operatorname{ho(co)lim}\alpha(X_\bullet).$
Relate the subquotients of $X_\bullet$ to the subquotients of $\alpha X_\bullet.$
Relate the subquotients of $\alpha X_\bullet$ to $\pi_2\operatorname{ho(co)lim} \alpha(X_\bullet).$

Step one works if $\alpha$ commutes with filtered homotopy limits or colimits (then take the filtration or cofiltration as appropriate). I don't know good conditions for this, but it seems to be common. Certainly (?) homotopy (co)limits commute with homotopy (co)limits, so we can get the spectral sequence of a homotopy (co)limit in this language. Also mapping spaces commute appropriately, so we can get the AH-SS.
Step two really is the input to this entire game. Note that $\alpha$ commutes with finite homotopy (co)limits (i.e. cones) so this is reasonable.
Step three is the spectral sequence of a filtered homotopy type. There is always a spectral sequence, but it may or may not converge to the groups of the homotopy (co)limit (depending on exactness properties of homotopy (co)limits on $A_2$).
I think many spectral sequences can be looked at in this way (certainly Grothendieck, AH, homotopy limit/colimit; probably also Adams).
To expand on Ben Wieland's example of the AH-SS, in this case $D_1 = D_2 = SH,$ $\pi_1 = \pi_2 = \pi$ is the stable homotopy groups, corresponding to the natural t structure, with heart abelian groups, $X$ is a fixed spectrum, $\alpha(E) = Map(X, E).$ Then $\alpha$ sends homotopy limits to homotopy limits, so we should use the filtration corresponding to $\pi,$ writing $E$ as the homotopy limit of its Postnikov filtration. The subquotients are Eilenberg-Maclane spectra corresponding to the homotopy groups of $E,$ and so we get a spectral sequence
$$ E_2^{*,*} = \pi_*\alpha(\text{subquotients of }E_*) = \pi_* Map(X, H\pi_*E) = H^*(X, \pi_*E) \Rightarrow \pi_*\alpha(E) = H^*(X, E)$$
which is what we wanted.