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Here are some things which are almost, but not quite, [representable by] (co)homology theories:

  1. $n$-excisive functors (in the sense of Goodwillie calculus), for $n \geq 2$.

  2. The domains and codomains of secondary, tertiary, etc. cohomology operations.

  3. The $E^r$ pages of various spectral sequences, for fixed $r$.

Now, (3) gives examples of (2), since the differentials of a spectral sequence are typically higher-order cohomology operations. I'm not sure to what extent (2) always arises from (3). But what I'm most interested in is the relationship between (1) and (2/3).

Questions:

  1. To what extent do higher-order cohomology operations necessarily arise as the differentials for some spectral sequence?

  2. Are the homotopy groups of the values of a 2-exisive functor, say, canonically defined using primary cohomology operations?

  3. "Conversely," is the $E^r$ page, of the Atiyah-Hirzebruch spectral sequence, say, for a fixed spectrum $F$, represented as the homotopy groups of the output of some $r+1$-excisive functor?

  4. Or is there some other relationship between (1,2,3) that I'm missing?

I suspect that a positive answer to (1) might come from the Goodwillie spectral sequence.

(My apologies for having two numbered lists!)

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  • $\begingroup$ Can you unpack a bit the statement that $n$-excisive functors are almost representable by cohomology theories? What do you mean? Regarding Q2, maybe start with the more basic question whether the $E^r$ page can be realized as the homotopy groups of any homotopy functor at all. My initial hunch was that the answer is no, but maybe Neil's answer shows a way to do it. But I would say that if $E^r$ is the homotopy groups of some homotopy functor, then it is a linear functor for all $r$ (true for $r=1, \infty$). Again, you can see a hint of this in Neil's answer. This ties in with my first point. $\endgroup$ Mar 9 '21 at 14:51
  • $\begingroup$ @GregoryArone Obviously you know this stuff a lot better than I do, but I have a few things in mind. Let's assume our functors are pointed and that our fuctors are from spectra to spectra for simplicity. Then of course, the "evaluate at $S^0$" map $Exc^1(Sp,Sp) \to Sp$ is an equivalence. So (i) to the extent that $n$-excisive functors are "generalizations" of 1-excisive ones, they "are" "generalized spectra". And (ii) the layers of the Goodwillie tower are represented by spectra with $\Sigma_n$ action, so $n$-excisive functors are built from spectral pieces. $\endgroup$
    – Tim Campion
    Mar 9 '21 at 15:00
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    $\begingroup$ You may get more examples from unstable SSs. E.g. I've been told the lower central series of a simplicial group fits into an analogy ($\mathbb{S}$ : $H\mathbb{Z}$) :: (Goodwillie tower : LCS) (e.g. Biedermann-Dwyer - homotopy nilpotent groups). The spectral sequence this gives was studied by Curtis in the 60s, and led to the better-known mod $p$ version, which is an unstable Adams SS (Bousfield-Curtis-Kan-Quillen-Rector-Schlesinger). Perhaps this suggests higher order excision is related to higher order unstable operations for stable co/homology theories? (This is just reckless speculation). $\endgroup$ Mar 9 '21 at 16:08
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This is not really an answer but it is an explanation of a point of view that I'd like to advertise. Let $\mathcal{C}$ be a triangulated category. Freyd constructed an abelian category $\mathcal{A}$ containing $\mathcal{C}$, as follows. For each morphism $u$ in $\mathcal{C}$ we have an object of $\mathcal{A}$, which I'll call $I(u)$. Given another morphism $v$, let $\text{Hom}(u,v)$ be the set of morphisms in the usual arrow category, i.e. pairs $(f,g)$ with $vf=gu$. Let $\text{Hom}_0(u,v)$ be the subset where $vf=gu=0$, and put $\mathcal{A}(I(u),I(v))=\text{Hom}(u,v)/\text{Hom}_0(u,v)$. For $X\in\mathcal{C}$ put $J(X)=I(1_X)\in\mathcal{A}$. Then one can check that $\mathcal{A}$ is abelian, and $J$ is a full and faithful embedding, and the objects in the essential image of $J$ are precisely the injective objects of $\mathcal{A}$, which are the same as the projective objects of $\mathcal{A}$. If $\mathcal{B}$ is another abelian category and $H\colon\mathcal{C}\to\mathcal{B}$ is a homology theory then we can define an exact functor $H'\colon\mathcal{A}\to\mathcal{B}$ by $H'(I(u))=\text{image}(H(u))$, and then $H'J\simeq H$.

A secondary cohomology operation from $E^*(X)$ to $F^*(X)$ is just a natural map $\ker(E^*(X)\to U^*(X))\to\ker(F^*(X)\to V^*(X))$ for some morphisms $u\colon E\to U$ and $v\colon F\to V$ of spectra. These biject with morphisms $\ker(J(u))\to\ker(J(v))$ in $\mathcal{A}$. Thus, secondary cohomology operations are just morphisms in $\mathcal{A}$. The fact that $\mathcal{A}$ is abelian means that tertiary (and higher) operations between familiar cohomology theories can be rewritten as secondary operations between less familiar spectra. And of course primary operations are just morphisms between projective objects in $\mathcal{A}$. Thus, the whole story about higher-order operations is really about the structure of $\mathcal{A}$, and I think it would be a good project to rewrite the traditional literature to make that explicit.

You can also note that a filtered spectrum gives rise to a spectral sequence in the abelian category $\mathcal{A}$. Homology theories give exact functors from $\mathcal{A}$ to $\text{Ab}$, and we can apply these to get the usual spectral sequences of abelian groups.

To relate all this to the Goodwillie story, you would want to study symmetric monoidal properties of $\mathcal{A}$ (which I think is straightforward) and the interaction with the theory of stable derivators.

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  • $\begingroup$ Thanks, Neil -- I'd seen Freyd's construction in your work before but had not appreciated this point! In Phantom maps and homology theories, you and Dan Christensen show that Freyd's construction universally turns a triangulated category into an abelian one (in the sense that homological functors out of $\mathcal C$ are the same as exact functors out of $\mathcal A$) and you also give a variant construction which universally turns $\mathcal C$ into an AB5 category. Could you say how your variant construction fits into this picture? $\endgroup$
    – Tim Campion
    Mar 9 '21 at 13:19
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    $\begingroup$ I don't have a strong opinion about whether Freyd's original version or the AB5 version is better. I suspect that there is no difference in any of the contexts where secondary operations have been used classically. $\endgroup$ Mar 9 '21 at 13:23
  • $\begingroup$ (For the record, Neil's construction with Dan is: from a nice tensor triangulated category $\mathcal C$ they define $\mathcal A'$ to be the category of additive functors $\mathcal F \to Ab$ where $\mathcal F \subseteq \mathcal C$ is a suitable subcategory of "finite" objects -- if I'm reading things right, they restrict to a setting where all possible notions of "finite" coincide -- compact, dualizable, being in the thick subcategory generated by the unit -- note that the embedding $\mathcal C \to \mathcal A$ in this case depends on having a monoidal structure). $\endgroup$
    – Tim Campion
    Mar 9 '21 at 13:33
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    $\begingroup$ It's confusing that neither the Freyd construction nor the Christensen-Strickland construction depend on the triangulated structure of $\mathcal C$ at all (whereas the notion of "homological functor" does)... $\endgroup$
    – Tim Campion
    Mar 9 '21 at 17:18
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This probably falls short of an answer, more like a rambling sequence of comments.

Regarding question 2, notice that $E^r$ is an additive functor. I.e., $E^r(X\vee Y)\cong E^r(X)\oplus E^r(Y)$. This implies that if $E^r(X)$ is realized as the homotopy groups of an $n$-excisive functor for some $n$, then it would be a linear functor, i.e, $n$ would be 1. But in this case $E^r(X)$ would be a generalized homology theory, which (unless I am very confused) it is not for $r>2$. So the answer to question 2 is No.

The moral that I would draw from this is that while there may be interesting connections between Goodwillie towers of specific functors and higher cohomology operations, a very general connection between $r$-excisive functors and higher cohomology operations or higher pages in a spectral sequence is perhaps too much to expect. In particular, the Atiyah-Hirzebruch spectral sequence does not appear to have much to do with higher excision.


Added later: A particular example of a Goodwillie tower that obviously has a lot to do with homology operations is the tower of the functor from spectra to spectra $X\mapsto H \wedge \Omega^\infty X$, where $H$ is the spectrum representing your favorite homology theory. The tower of this functor calculates the homology of $\Omega^\infty X$ in terms of the homology of $X$. The differentials have a lot to do with homology operations, specifically Dyer-Lashof operations. All this was studied in considerable detail (for $H=H\mathbb Z/2$) in the following paper of N. Kuhn and J. McCarty

The mod 2 homology of infinite loopspaces, Algebr. Geom. Topol. 13 (2013)


Regarding question 1, you could say that the answer is yes, for trivial reasons, if you interpret broadly enough what it means to be "defined using primary cohomology operations". A $2$-excisive functor is the homotopy fiber of a map between two spectra, so if any map between spectra is a cohomology operation, then a functor is determined by a cohomology operation.

To be more specific and a little less trivial, let me recall a general description of $2$-excisive functors. We restrict ourselves to pointed functors between the categories of pointed spaces or spectra. Then any quadratic functor is the homotopy fiber of a natural transformation of the following form, where $C$ is a spectrum and $D$ is a spectrum with an action of $\Sigma_2$: $$ C\wedge X\to (D\wedge X^{\wedge 2})_{h\Sigma_2} $$ (in the case of space-valued functors you should put $\Omega^\infty$ in front of these functors, and there may be non-deloopable natural transformation. These correspond to the unstable homotopy operations that were mentioned in a comment).

Natural transformations of this form are well-understood, and can be described in terms of maps from $C$ to (some spectrum constructed out of) $D$. If you call these maps cohomology operations, then you can say that $2$-excisive functors with prescribed first and second layers are classified by cohomology operations of certain type from the first layer to the second.

In case of functors from/to the category of spaces, the derivatives have the structure of a module over the spectral Lie operad. There is a relationship between this module structure and the differentials in the Goodwillie spectral sequence. So you can say that to some extent the cohomology operations that determine the differentials in the Goodwillie spectral sequence come from this Lie module structure. But this is not the whole story.

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  • $\begingroup$ Thanks! Already your basic observation that $n$-excisive functors generally don't preserve sums goes a long way toward showing that the kind of correspondence I was hoping for is nonsense. Is it precisely the case that an $n$-excisive functor which preserves sums must be 1-excisive? $\endgroup$
    – Tim Campion
    Mar 9 '21 at 18:05
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    $\begingroup$ Yes I believe so. $\endgroup$ Mar 9 '21 at 18:07

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