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I’ve been working on spectral sequence. But not done yet.
I need to see what actually the fully pecise assumptions for the various theorems are.
Can anyone point me to a good comprehensive reference that preferably uses the full model theory category tools throughout? Large bits of the homological algebra literature still have not arrived in the 21st century…
I fixed some typos.
Can anyone point me to a good comprehensive reference that preferably uses the full model theory category tools throughout? Large bits of the homological algebra literature still have not arrived in the 21st century…
While I am eventually sympathetic toward your complain, be aware that you are complaing why not model categories (created at the end of 1960s) while the entry is at the moment not even up to earlier discovery of triangulated categories (devised in early 60s). Thus the entry is in 1950-s, while the books from the end of 20th century like the Gelfand-Manin methods book treat the subject from the perspective of total derived functors among triangulated categories, rather than each cohomology group separately. Grothendieck’s spectral sequence (discovered around 1955 in old language) is just a rewording of the composition rule for composite derived functor and it is in Gelfand-Manin Methods phrased in terms of adapted classes, hence it does not need very restrictive things like enough injectives; it is natural to treat it for triangualted categories I have unfortunately no luxury to help you this week (though it would be a pleasant and rewarding task), but if I were to modernize the entry, there are (more substantial) levels of modernizing even before going to 21 century. It would be also interesting to know what one can say from the alternative language of derivateurs.
A reason more to look in triangulated context is that the “stable” category of spectra is triangulated so really a lot of examples are nicely treated in this context. P.S. But the treatment of filtered objects for stable (infinity,1)-categories in Lurie’s “stable” paper is not sufficient for your 21st century purposes ?
I am trying to bring spectral sequence into shape, eventually. So far I have mainly rewritten/expanded the Idea-section.
I have started organizing the List of examples of spectral sequences by special-case-relation: classes of spectral sequences further to the right are special cases of those further to the left.
Not done yet, but I’ll stop doing this for the moment.
I have added more details to Definition - Convergence (mentioned “limit term” and “abutment”, which weren’t mentioned before). Also added Definition - Boundedness together with the argument that a bounded spectral sequence indeed has a limit term.
At spectral sequence in the Idea-section it used to say “Despite their name, there is nothing specifically spectral about spectral sequences, for none of the …” and so on
I have changed that to read:
Despite their name, there seemed to be nothing specifically “spectral” about spectral sequences, for none of the technical meanings of the word spectrum. Together with the concept, this term was introduced by Jean Leray and has long become standard, but was never really motivated (see p. 5 of Chow). But then, by lucky coincidence it turns out in the refined context of stable (∞,1)-category theory/stable homotopy theory that spectral sequences are essentially the incarnation on homotopy groups of sequences of spectra. This is discussed at spectral sequence of a filtered stable homotopy type.
Hmm. You aren’t claiming that all spectral sequences are the incarnation on homotopy groups of a sequence of spectra, are you?
No, I am not claiming anything but what is evident in the literature.
In practice there are already few spectral sequences which are not special cases of the spectral sequence of a filtered complex. The notion of spectral sequence of filtered stable homotopy types is more general, hence makes this traditional statement even more true.
Okay, then I changed “are essentially the incarnation on” to “frequently arise by considering the”.
Incidentally: what are examples in practice that are not of this kind?
Well, I don’t know offhand how to obtain the homotopy spectral sequence of a tower of spaces by passing through spectra, since it involves unstable homotopy groups.
No wait, this is abuse of the red herring principle. A homotopy spectral sequence is not a spectral sequence.
What examples do we have of actual spectral sequences arising in practice which don’t come down to being spectral sequences of a filtered complex or more generally of a filtered stable homotopy type? Any?
Whaa? What does “homotopy spectral sequence” mean to you? I just meant the spectral sequence whose first page is built out of the homotopy groups of the spaces in a tower. I think one says “homotopy spectral sequence” to distinguish it from a “homology spectral sequence” or a “cohomology spectral sequence” whose first pages are homology or cohomology groups.
Yes, so since a homotopy spectral sequence is modeled on homotopy groups of a topological space, it contains nonabelian groups and pointed sets in low degree, instead of abelian groups or else objects of an abelian category, as the definition of spectral sequence requires.
E.g.
Ok, it’s true that when people give a precise mathematical definition of the phrase “spectral sequence”, they tend to give only the abelian version. But informally, the subject of spectral sequences includes also nonabelian ones, so I think it would be disingenuous to make a sweeping general statement about “spectral sequences” without qualification if it only applies to the abelian ones.
More directly, if you happen to build the homotopy spectral sequence of a tower of simply connected spaces, it will consist only of abelian groups, yet I still don’t see how to obtain it from spectra.
@Mike
But they are only abelian groups after applying the forgetful functor from graded Lie rings (not sure I do agree entirely with Urs, but just to play devils advocate).
Okay, I guess I should have said “types of spectral sequences” instead of just “spectral sequences”.
Concerning sweeping statements: we had analogous discussions before, and thanks for the advice, but allow me to offer the perspective from my side: often I feel like I am saying
“The sky is blue.”
only then to draw flak that feels like
“No! At night it’s black.”
In the present context I took the stability for granted and was thinking it’s the filtering aspect that deserves to be highlighted. But, true, there is also an unstable context. It’s still controled by filtering, though.
For people proposing a formal system which makes such a big deal of context, should we be better at detecting implicit context, or fussier that it’s not explicit?
David #21: I don’t really grasp the point of the question (and particularly I don’t know if that ’or’ is exclusive), but when it comes to presenting material that others will scrutinize, I think it’s better to be fussy about presenting the contextual details.
Todd, it wasn’t meant very seriously. In mathematics specification of context is, of course, vitally important. It’s less likely, then, that a formalism that codifies contexts explicitly will make much difference. But there is a more serious thought that I’m working on at the moment that there could be a significant effect concerning philosophy. Belief that untyped predicate logic does a good job at capturing, or that it even improves upon, reasoning in natural language seems misguided to me.
Once nouns and adjectives are codified via predicates and quantifiers, it gives rise to the idea that all grammatical sentences must have truth values. I prefer a system which makes me think about what must be in place before introducing terms, e.g., ’the King of France’.
Perhaps we even see a shadow of this in everyday life when a journalist or lawyer demands a Yes or No answer to a question. People may have some sense of the failings of reasoning brought to light by deductive logic - false dichotomy, affirming the consequent, etc. But perhaps it’s time that they also knew about faulty term introduction.
A similar presupposition in philosophy operates in the Bayesian framework where people say things like “Degrees of rational belief form a mapping from the sentences of a language to the interval [0, 1], such that…” It might be interesting to think of Bayesianism operating in a type theoretic framework.
A spectral sequence should be allowed to take values in any category $\mathcal{H}$ in which it makes sense to ask that $H(E_r,d_r)\simeq E_{r+1}$, and maybe where the notion of exact couple also makes sense. As Marco Grandis shows, it suffices that $\mathcal{H}$ be a homological category. What would be nice is to generalize the spectral sequence of a filtered stable homotopy type to this context. Namely, one should be able to define homological and cohomological functors from a (∞,1)-category $\mathcal{C}$ to a homological category $\mathcal{H}$, and then associate to any filtered/cofiltered object in $\mathcal{C}$ a spectral sequence in $\mathcal{H}$. Then I might believe the claim that every relevant spectral sequence arises in this way.
Perhaps a problem with this claim is that the above construction only produces bigraded spectral sequences. Are there any genuinely non-bigraded spectral sequences out there? The usual example is the Bockstein spectral sequence, which is monograded, but that’s because it arises from a single-step filtration.
Mike, Marc, and others who feel that spectral sequences should be understood more generally than in abelian categories:
if you really feel strongly about this then you might want to go and edit the nLab entry spectral sequence accordingly. Because that entry assumes an ambient abelian category (and always did so, not just since I made that last edit which spawned this discussion here), just like the corresponding entries all over the web do, e.g. on Wikipedia, on PlanetMath and on the the Springer Online Encyclopedia of Mathematics, do.
In fact, as far as I can see, the nLab stands out among these references as being the only place that mentions that there is a generalization of the notion of spectral sequences to homotopy spectral sequences.
But if you feel that this is not a generalization but is the default, then maybe it’s worth editing the nLab entry accordingly.
@Urs I will edit the entry if I can work out something along the lines of post 24. I’m not interested in just including an ad hoc or vague definition of “homotopy spectral sequence”.
I think it’s fine for the page spectral sequence to be about the abelian version with a more general definition at homotopy spectral sequence.
To come back to the question in #13: What are examples of spectral sequences that do not arise from a filtering?
Over on MO Peter May says that the Bockstein spectral sequence is an example.
But it seems to me that in
it is claimed and discussed that the Bockstein SS can be realized as a filtration SS after all.
I have expanded the statement of the definition of convergence of a spectral sequences by distinguishing the cases “weak convergence”, “convergence” and “strong convergence”, as in Boardman 99.
After the subsection on convergence, I have added a subsection The extension problem with some basic remarks.
Added:
Spectral sequences were originally introduced in 1946 by Jean Leray in the paper
The nLab has another 1946 Leray CR paper uploaded, L’anneau d’homologie d’une représentation. It seems it’s from the week before (published I think on the 12th June), and is only about 50 pages earlier in the journal. You can almost see the idea there, but the paper in #33 (published on the 17th June, pdf) has the real deal, it seems. The paper from the 17th cites the one from the 12th, the first of its three citations.
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