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stub for Goodwillie calculus
I added few and compacted the references.
Perhaps a good article to extract some material - Introduction to the manifold calculus of Goodwillie-Weiss.
No time to read it, but looking forward to whatever you can extract from it for our entry Goodwillie calculus. (Added at least the reference to the References-section.)
I filled a good part of Andre Joyal’s latest message to the Category Theory Mailing list into a section Analogy between homotopy theory and caluclus at Goodwillie calculus.
This could do with more editorial polishing, but I am too tired now.
Funny…I haven’t got Andre’s email yet. I think my categories list is slow. :)
I added some blog material and quotes from ’The Goodwillie tower of the identity is a logarithm’.
Ah, thanks! Very interesting.
I wonder if Andre Joyal is still following the forum discussion here. I would be interested in hearing his opinion on this.
Added a couple of motivating ideas by Arone and Finster.
I keep thinking the Goodwillie calculus should ultimately be very simple and sensible, but I’ve never met anyone who has seen through it to the bottom and can explain it to me nicely. I think Joyal understands it, but I’ve never gotten a chance to quiz him about it.
@John:
Tom Goodwillie is on MO, so if you ask a question about this over there, I’m sure he’d be able to respond.
Unfortunately, reading Goodwillie’s papers does not suggest that he understands the Goodwillie calculus in the particular way that André Joyal understands it, and only Joyal has ever given me the feeling that his explanation would make me happy. We already have a link to Joyal’s remarks on the category theory mailing list, but I would like him to expand on these!
I wonder if my hunch was right. That might prove to be a way in if we could understand how things like finite difference calculus relate to the cosmic cube. I dare say homotopic species are around some place too.
Make your hunch more precise, so I can tell more easily if it’s right.
(Yes, this is supposed to be one of those Socratic anamnesis tricks, except I don’t know the answer either.)
Concerning the meaning of Goodwillie calculus:
If find Joyal’s description a nice unwrapping, but to my mind the nicest abstract description of what’s going on is Jacob Lurie’s.
Just the intro of his section 5 tells me exactly what the Goodwillie calculus is, and why it is. The rest is details.
Unfortunately that passage by Goodwillie doesn’t help me - at least, no more than Goodwillie’s papers. It makes just enough sense to let me know that ultimately, the Goodwillie calculus is something very simple, very much just a categorified version of the theory of Taylor series.
For an example of why I find most descriptions of the Goodwillie calculus frustrating, consider this paper title:
’The Goodwillie tower of the identity is a logarithm’.
Now, the Taylor series of the identity should not be a logarithm! The Taylor series of the identity should be
$x$The Taylor series of the logarithm should be a logarithm. This suggests that the Goodwillie calculus as typically formulated involves a change of variable that sense multiplication to addition. And that’s clear from accounts that say the simplest kind of functor - from the Goodwillie point of view - is one that sends limits to colimits.
So one of my questions has always been whether one can ’factor’ the Goodwillie calculus into an ’easy’ part and a ’change of variables’ part. Hmm, maybe Lurie’s commutative squares at the very beginning of section 5 address this issue.
But I guess I’ll have to reread Joyal’s remarks and think about them a lot… it would be so much quicker if I could talk to him!
@John: You could always email him, no?
I understand why it would be nice to have an analogy between Goodwillie caluclus and Taylor expansion. But I don’t see why that would serve more to achieve a description that is
ultimately be very simple and sensible, but I’ve never met anyone who has seen through it to the bottom and can explain it to me nicely.
than the nice abstract category theory that Lurie presents.
I have a similar remark about Andre Joyal’s talk about the Dold-Kan correspondence, which in a similar vein he related to arithmetic:
while I see why it is nice to have an analogy between the Dold-Kan map and Newton’s finite differences, I by no way find this more insightful than the equivalent category-theoretic statement, that we are homming simplices into a complex. Quite on the contrary: I find the description in terms of simplices nice and insightful, and the equivalent description in terms of finite difference calculus a curious side effect. For me this gives a nice way to organize the finite difference calculus conceptually in terms of simplices instead of organizing the Dold-Kan correspondence nicely in terms of finite difference calculus.
For Goodwillie calculus I have similar feelings (only that I haven’t spend much time thinking about it). What could be nicer and more insightful than Lurie’s description? But of course that’s just me.
So do the motivational comments on the page fall under one of (a) clear once you understand Lurie’s abstract approach, or (b) belong to some vague analogy with things like finite differences and Taylor series? If so, perhaps the first of the following is (b) and the second (a)?
The functor from spaces to spaces which sends $X$ to
$\Omega^{\infty}\Sigma^{\infty} X = colim \Omega^n \Sigma^n X$sends coproducts to products and is supposed to be like $e^{x - 1}$. (The “$-1$” comes about from issues to do with basepoints.)
and
…the category of spectra plays the role of the tangent space to the category of spaces at the one-point space. Moreover, the identity functor from spaces to spaces is not linear…and one can interpret this as saying that spaces have some kind of non-trivial curvature.
So do the motivational comments on the page fall under one of (a) clear once you understand Lurie’s abstract approach, or (b) belong to some vague analogy with things like finite differences and Taylor series?
I am not claiming that it is clear how the Taylor expansion-discussion follows from the abstract category theoretic description. And I am not saying that it is not highly interesting to understand it. I was just making a comment on what I think of as “getting to the bottom”. It’s also not an important remark. You are all invited to ignore it!
the category of spectra plays the role of the tangent space to the category of spaces at the one-point space.
By the way, this statement is true as a precise statement in terms of Lurie’s insights: the tangent (infinity,1)-category of the $(\infty,1)$-category $\infty Grpd$ over $*$ is the stabilization of the over-category $\infty Grpd /*$. That’s precisely the stable $(\infty,1)$-category of spectra.
But I don’t see why that would serve more to achieve a description that is … than the nice abstract category theory that Lurie presents.
My own personal reaction (which might not be anyone else’s, particularly John’s) is that that nice abstract category theory description – and nice it is! – foreshadows a long uphill quasi-categorical climb in order to understand and make use of. For those not yet fluent in quasi-categories – and this includes me – that abstract description (as given in the nLab) may look somewhat forbidding and remote, and the analogy with calculus is really not that obvious unless one is immersed in the subject.
To me, the analogies proposed by Joyal look like a far less intimidating key of entry to answering the question, “What is the Goodwillie calculus, and what is it good for?” No doubt that making contact with Lurie’s approach is important at some point for those who really wish to understand, but on first approach, Joyal’s may appear rather more inviting.
I am submitting this comment with some hesitation, as I’m worried it will incite an argument with Urs (which I don’t want, or even have much time for at the moment).
For those not yet fluent in quasi-categories – and this includes me
I get the impression that practically nobody is fluent in all of the foundational material (except a select few experts).
I like both Joyal’s description and Lurie’s. Lurie’s is a nice description of why you might be interested in this thing if you already care about stabilizations, but it doesn’t seem to say anything about in what sense this is “like calculus,” except insofar as it involves approximation of one thing by something else. On the other hand, Joyal’s description is compelling as to why we call this “calculus”, but it doesn’t seem to go beyond intuitive analogies yet. It seems to me that the two descriptions must be just two sides of the same coin, if we could just figure out how to describe the whole coin. The Arone-Kankaanrinta proposal that Goodwillie calculus happens in “log space” seems particularly promising to me; as John said, the fact that the derivatives of the identity functor are highly nontrivial has always been a significant barrier for me to believe any analogy with ordinary calculus.
The reason why I like Joyal’s analogy is that, following in the footsteps of his work on species, I’ve spent a lot of time working on categorified arithmetic, algebra, calculus, and so on. The goal is to take all of elementary mathematics and see it as a decategorified, watered-down version of something truly beautiful. Once it’s working well enough, we should be able to take any of our favorite high-school trig identities, or formula for integrals, and see that it has a deeper meaning in terms of $\infty$-categories - or homotopy theory, if you prefer. This is already possible in a vast number of cases, some of which are explained here and here. But there are still many difficulties left to work out.
(In fact, “week300” will about this stuff. We can categorify the Riemann zeta function!)
Joyal’s species categorify the concept of ’formal power series’ in a very nice and very fruitful way, with a nice relation to ’analytic functors’. The Goodwillie calculus seems to be about extracting a Taylor series approximation to a fairly large class of functors. So you’d think it would be closely related to Joyal’s work… but I don’t see exactly how, and that’s what’s bugging me.
Harry Gindi wrote:
@John: You could always email him, no?
Sure - but if you’re trying to get someone to explain something, conversation works a lot better. You can do stuff like go “huh?” in the middle of someone’s sentence. And it’s really fun when Joyal explains stuff. I’ve had various chances to ask him questions, and I hope to get some more someday…
Now we know that
the category of spectra plays the role of the tangent space to the category of spaces at the one-point space,
how can we understand:
The functor from spaces to spaces which sends $X$ to
$\Omega^{\infty}\Sigma^{\infty} X = colim \Omega^n \Sigma^n X$sends coproducts to products and is supposed to be like $e^{x - 1}$?
The first thing that comes to mind with tangent spaces and exponentials is the Lie group/Lie algebra relationship, but maybe that doesn’t help. Can Joyal’s analogy be pushed further? If $k[[x]]$ corresponds to the category of pointed homotopy types, what corresponds to its tangent category at the one element pointed set?
So, does anybody have notes from Joyal’s talk in Categories conference in Genova, to post a scan ? There was supposedly much more there than it is in the categories-list post. Leinster ? Berger ?
Yeah, with all the talk of this mysterious Joyal seminar/interpretation even I’m curious now :)
There was supposedly much more there than it is in the categories-list post. Leinster ? Berger ?
As I suppose you have seen, Tom seemed to have recounted of the talk what he was willing to recount in public on the $n$Café in the Dold-Kan thread (see also one comment of his further below).
He just mentioned it, did not give us the scan of his notes.
Does the following by Goodwillie chime with anything done at nLab, I mean the use of differential geometry language in homotopy theory:
Rhetorical question: If the first derivative of the identity is the identity matrix, why is the second derivative not zero? Answer: Some of the terminology of homotopy calculus works better for functors from spaces to spectra than for functors from spaces to spaces. Specifically, since “linearity” means taking pushout squares to pullback squares, the identity functor is not linear and the composition of two linear functors is not linear.
Attempted cryptic remark: Unlike the category of spectra, where pushouts are the same as pullbacks, the category of spaces may be thought of has having nonzero curvature.
Correction: After the talk Boekstedt asked about that remark. We discussed the matter at length and found more than one connection on the category of spaces, but none that was not flat. In fact curvature is the wrong thing to look for. There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed.
This is from the report (p. 905) on a Oberwolfach meeting. The table on p. 900 also makes comparisons to differential geometry.
Thanks Eric. I’ve included a link to the post at the nLab page. Of course, feel free to work on that page. It’s quite a mess at the moment.
added pointer to
Saul Glasman, Stratified categories, geometric fixed points and a generalized Arone-Ching theorem (arXiv:1507.01976, talk notes pdf)
Saul Glasman, Goodwillie calculus and Mackey functors (arXiv:1610.03127)
for discussion of Goodwillie calculus via spectral Mackey functors
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