origin of spectral sequences in algebraic topology

Question

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

Answer 1

It seems to me that everything written so far is addressing the title, not the body of the question, in particular focusing on the unstable. Part of that is people asking their own questions. But I think it is best to start with the easiest questions. Indeed, Tom, you have singled out the right place to start: Yes, the Atiyah-Hirzebruch spectral sequence is a hypercohomology spectral sequence.

Of course, we need a generalization or abstraction of the hypercohomology spectral sequence to make sense of that. Surely, you know that as you move from homological algebra to stable homotopy theory, you should generalize from derived categories of abelian categories to the abstraction of triangulated categories.* But there is another abstraction that you should know about, that of t-structures.

As Dylan says, spectral sequences come from filtrations.** But where do filtrations come from? One source is taking a chain complex and filtering by degree. The subquotients reveal exactly the original chain complex, while we'd prefer something that depended only on the quasi-isomorphism class. (Though often this is good enough because, although its $E_1$ page is not canonical, the rest of the sequence is.) The steps of this filtration are called the "naive truncations." They have the property that their homology agrees with the original in low degrees, is zero in high degrees, and is not canonical in a single degree. With a little modification, this can be changed into the good truncation, which has no such intermediate degree, but goes directly from agreeing to disagreeing.

Here is one version of the hypercohomology spectral sequence, I think pretty close to way that you see it. Start with a right-derivable functor $F\colon A\to B$ between abelian categories and the goal of understanding its derived functor on chain complexes $RF\colon DA\to DB$. Start with a complex $C\in DA$, take its good filtration, so that its subquotients are $H^iC[i]$, shifts of objects of $A$, and apply $RF$ to get a chain complex made of $R^jF(H^iC)$. Then as you reassemble $C$ from the $H^iC$, the spectral sequence reassembles $RF(C)$ from $RF(H^iC)$.

Let us abstract what this argument required: not that $DA$ was the derived category of an abelian category, but merely that each object had a filtration whose subquotients lived in shifts of an abelian category, called the heart; and that it was easier to understand the functor restricted to this abelian category. The first condition, which does not mention the functor, is called a t-structure.

Ultimately, the point is that the stable homotopy category has a t-structure whose heart is the category of abelian groups, but exotic t-structures abound and the concept was introduced in algebraic geometry. The easiest example is given by duality: The derived category of perfect chain complexes of abelian groups (that is, bounded complexes of finitely generated free abelian groups) is equivalent to its opposite under the contravariant duality functor $\mathrm{Hom}(-,\mathbb Z)$. Thus, we can transport the t-structure across the duality to get a new t-structure on the old category. The old heart is the category of finitely generated abelian groups (placed in degree zero). The new heart is equivalent to the opposite of the old heart. It consists of objects that are the sum of a free abelian group in degree zero and a torsion group shifted by $1$.

The filtration from the t-structure on the stable homotopy category is called the Postnikov filtration. The objects of its heart are called the Eilenberg-MacLane spectra; their associated cohomology theories are usual cohomology. Thus if our functor of interest is cohomology of a fixed space $X$ with coefficients in the varying spectrum $E$, its restriction to the heart is ordinary cohomology, a well-understood starting point, and so the hypercohomology spectral sequence is $H^i(X;E^j)\Rightarrow E^{i+j}(X)$, the same form as the Atiyah-Hirzebruch spectral sequence. (You may have to do some work to check that it is the actually the same spectral sequence.)

* Probably what I have to say can be made to work under Verdier's axioms for triangulated categories, but I really mean stable $\infty$-categories. Or you could work with DG-categories, until you want to move to stable homotopy theory.

** The Bockstein spectral sequence is a purely algebraic spectral sequence that I do not know how to see as coming from a filtration, though I have a vague memory of another spectral sequence that does come from a filtration and carries the same information.

Answer 2

I am very surprised that nobody has yet mentioned Massey's beautiful general theory of exact couples. This to my mind answers the alternative version of the question, and does so without restriction to algebraic topology. Filtrations give rise to exact couples, but they can perfectly well come from data that has nothing to do with any filtration of anything. A beautiful, important, and breathtakingly elementary example is the Bockstein spectral sequence of a chain complex, which tells us how to obtain $p$-local information from mod $p$ homology. It is the only example I know of a spectral sequence which is singly graded, and its mere existence illustrates why one should not think solely in terms of filtrations. Exact couples are much more elementary and much less tied to context than most starting points for the study of spectral sequences.

Answer 3

I'm not comfortable enough with spectral sequences to answer this question, but let me answer an easier version of this question with spectral sequences replaced by long exact sequences.

In algebraic geometry I think all of the long exact sequences you'll run into ultimately come from deriving some functor between abelian categories. In algebraic topology, on the other hand, a very important example of a long exact sequence is the long exact sequence of a fibration, which ought to come from the "derived functor of $\pi_0$" but can't possibly be obtained from the usual abelian category story because in general it involves nonabelian groups.

There are two ways to go from here (that I know of). One is to broaden your notion of derived functor to include nonabelian examples. I'm not comfortable enough with this story to explain it in detail, and in particular I haven't checked that the details work out, but the basic idea is to replace resolutions involving (co)chain complexes with resolutions involving (co)simplicial objects (the Dold-Kan correspondence tells you that the latter reduces to the former in an abelian category, which is some evidence that this is a good idea).

But there is a more directly topological story: the long exact sequence of a fibration reflects a more fundamental underlying structure, namely that of fiber sequences. Start with a pointed continuous map $f : E \to B$ between pointed spaces. Then we can construct the homotopy fiber $F$ of this map, which (at least after fibrant replacement) gives a fibration

$$F \to E \to B$$

of pointed spaces. Now the curious thing about taking homotopy fibers is that the homotopy fiber of a homotopy fiber, unlike say the kernel of a kernel, is in general nontrivial: if we now take the homotopy fiber of $F \to E$, we find from a standard lemma about homotopy pullbacks (which takes the same form as the corresponding lemma about ordinary pullbacks) that this is the same as the homotopy fiber of the inclusion $\bullet \to B$ of the basepoint into $B$. But this is precisely the based loop space $\Omega B$! Continuing to take homotopy fibers in this way, we get a sequence of fibrations

$$\dots \to \Omega^2 F \to \Omega^2 E \to \Omega^2 B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B$$

and applying $\pi_0$ to this sequence gives the long exact sequence of homotopy groups, using the fact that mapping into spaces behaves well with respect to homotopy limits. (There is a dual story about how to get long exact sequences in homology and cohomology from cofiber sequences which involves repeatedly suspending rather than taking loop spaces.)

This is in some sense the nonabelian version of the long exact sequence associated to a short exact sequence of chain complexes, and can be run in any setting where you have a sufficiently well-behaved notion of homotopy limit.

(To get from here to, say, the spectral sequence of a homotopy limit I have been told that the idea is to start by defining a suitable filtration of the homotopy limit. After fibrant replacement this gives a tower of fibrations, and I think patching together the long exact sequences of these fibrations in some way should give the spectral sequence. I'm sure an expert can say more here, though. This should be the nonabelian version of the spectral sequence associated to a filtration of a chain complex.)

Answer 4

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:

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

2. Relate the subquotients of $X_\bullet$ to the subquotients of $\alpha X_\bullet.$

3. 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.