2308.15573 is an exciting development, especially the characterization of topoi of n-excisive functors as classifiers for “n-nilpotent objects”. I’m still digesting the technicalities of nilpotence properties in this context and don’t feel confident adding anything to Goodwillie calculus or n-excisive (∞,1)-functor, but it feels important. Perhaps someone here is equipped and motivated to summarize and contextualize the result on the relevant lab pages?

]]>started an Examples-subsection “Goodwillie derivatives of the exponential modality” (here)

]]>added pointer to:

- Gregory Arone, Michael Ching,
*Goodwillie Calculus*, in:*Handbook of Homotopy Theory*Taylor &Francis (2019) [arXiv:1902.00803,doi:, 10.1201/9781351251624]

Previous edits fixed some reference bit-rot: namely, the Oberwolfach report and Eric Finster’s blog post. I removed a broken link (http://people.virginia.edu/~elf9e/research.pdf) to Eric Finster’s research statement (couldn’t find a web archive link to it).

]]>added pointer to:

- David Barnes, Rosona Eldred,
*Capturing Goodwillie’s Derivative*, Journal of Pure and Applied Algebra**220**1 (2016) 197-222 [arXiv:1406.0424, doi:10.1016/j.jpaa.2015.06.006]

Added

]]>Discussion in an equivariant setting is in

- Emanuele Dotto,
Higher Equivariant Excision, (arXiv:1507.01909)

have tried to clean up the list of references (re-ordered a little for more systematics, and fixed up some of the more neglected items in the list)

]]>added links to author and pdf of:

- Nicholas Kuhn,
*Goodwillie towers and chromatic homotopy: an overview*, Proceedings of the Nishida Fest (Kinosaki 2003), 245–279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007 (arXiv:math/0410342, doi:10.2140/gtm.2007.10.245)

added links and DOI-s to:

Thomas Goodwillie,

*Calculus. I. The first derivative of pseudoisotopy theory*, K-Theory**4**(1990), no. 1, 1-27 (doi:10.1007/BF00534191, MR:1076523)Thomas Goodwillie,

*Calculus. II. Analytic functors*, K-Theory**5**(1991/92), no. 4, 295-332 (doi:10.1007/BF00535644, MR:1162445)Thomas Goodwillie,

*Calculus. III. Taylor series*, Geom. Topol. 7 (2003), 645–711 (euclid:gt/1513883319 doi:10.2140/gt.2003.7.645, arXiv:math/0310481))

Fixed the pdf link and added full publication data to:

- Tom Goodwillie,
*The differential calculus of homotopy functors*, Proceedings of the International Congress of Mathematicians in Kyoto 1990, Vol. I, Math. Soc. Japan, 1991, pp. 621–6 (article pdf, full proceedings Vol I pdf)

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

]]>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.

]]>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.

]]>He just mentioned it, did not give us the scan of his notes.

]]>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).

]]>Yeah, with all the talk of this mysterious Joyal seminar/interpretation even I’m curious now :)

]]>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 ?

]]>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?

]]>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…

]]>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.

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).

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).

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