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The lack of resources bridging the gap between what one finds in Hatcher's algebraic topology text and modern research on homotopy theory has been brought several times before on MathOverflow [1, 2, 3].

Clark Barwick states [4], in “The Future of Homotopy Theory”

I believe that we should write better textbooks that train young people in the real enterprise of homotopy theory — the development of strategies to manipulate mathematical objects that carry an intrinsic concept of homotopy. These textbooks have the power to be useful not only for people at the beginning of their careers, but for a large swath of non-experts as well.

However, the work involved in writing a textbook/survey of homotopy theory is a herculean one, and having a “Homotopy Theory” project could perhaps be the best solution for this. Being collaboratively, it would not require researchers to set aside a tremendous amount of time in writing an entire textbook by themselves (or in small collaboration).

This question has two purposes. The first is to stir discussion (which would perhaps be most appropriately done in the Homotopy Theory chat). The second (which adheres to the Q/A format of MathOverflow) is to ask:

What would be the biggest hurdles in carrying such a project?

Another question$^*$: In the comments, Timothy Chow points that “[...] Perhaps a productive way forward would be to create a detailed sketch of such a backbone that a single person could plausibly write (and then, ideally, write it yourself, after getting some feedback).” So, what would such a sketch of the backbone be? That is, what topics form the gap between Hatcher and modern research?

[1] Algebraic Topology Beyond the Basics: Any Texts Bridging The Gap?

[2] Why not a Roadmap for Homotopy Theory and Spectra?

[3] Roadmap to Hill-Hopkins-Ravenel

[4] The Future of Homotopy Theory

$^*$Added in an attempt to point the question in an useful direction.

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Chromatic Homotopy Theory
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    Not exactly what you are looking for, but there is the effort of Haynes Miller's Handbook of Homotopy Theory. This is an on going project, and it seems that the material there is really good, although maybe not really introductory at times. This page has links to the parts already written: https://ncatlab.org/nlab/show/Handbook+of+Homotopy+Theory – Shay Ben Moshe Feb 10 '19 at 18:25
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    @ShayBenMoshe It's not the same thing at all. This handbook is a collection of chapters on different themes of homotopy theory that are independent from each others (the chapters, not the themes). It's completely different from the stacks projet, whose goal is to write a coherent textbook on a single theme. – Najib Idrissi Feb 10 '19 at 18:42
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    @NajibIdrissi I agree with every word you wrote. However I do think it was worth mentioning as a comment, because that book might be very useful for people interested in the suggested homotopy theory stacks project. Furthermore, it has the huge advantage that it actually exists! – Shay Ben Moshe Feb 10 '19 at 21:41
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    As Harry Gindi's answer (and the comments) makes clear, it is a misconception that a collaborative project eliminates the need for any single person to invest a tremendous amount of time. Once a "backbone" is in place, others can add to it without investing tremendous amounts of time, but the backbone will almost certainly need to be written by a single person. Perhaps a productive way forward would be to create a detailed sketch of such a backbone that a single person could plausibly write (and then, ideally, write it yourself, after getting some feedback). – Timothy Chow Feb 11 '19 at 00:31
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    To contrast, without supreme organisation and discipline, what you get is a more organic and idiosyncratic structure like the nLab, which isn't and never was meant to be anything like the Stacks Project. – David Roberts Feb 11 '19 at 05:54

2 Answers2

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Easy. Who's gonna write it?

JDJ (Johan de Jong) has written almost the entire stacks project himself.

Todd Trimble
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Harry Gindi
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    This reminds me: do you remember seeing a copy of some email by Kevin Buzzard (either brought up by de Jong or by Buzzard himself) in the days before the stacks project, sounding a cautionary note along these lines? – Yemon Choi Feb 10 '19 at 17:58
  • @YemonChoi Nope, I think the Stacks project was already over a thousand pages when I started my undergrad. – Harry Gindi Feb 10 '19 at 18:00
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    Jacob Lurie? https://www.math.columbia.edu/~dejong/wordpress/?p=4243 – Will Sawin Feb 10 '19 at 18:01
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    @WillSawin I'll let Jacob speak for himself, but I don't think that he's aiming to write a reference for all of homotopy theory, just higher categories in homotopy theory. The point my answer was making is that these aren't really community efforts, so community input is pointless. – Harry Gindi Feb 10 '19 at 18:04
  • Maybe some researchers could get together and write small parts of it (like the 50 page chapters of the Handbook of Homotopy Theory), and then that writing is improved over many years by different people, or the various existing lecture notes are incorporated into such a project and it is slowly rewritten to make it coherent. While this may seem childishly hopeful, I think it could be done: currently there are lots of homotopy theorists unhappy with the current expository state of the subject... – Chromatic Homotopy Theory Feb 10 '19 at 18:09
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    @ChromaticHomotopyTheory What you're describing is called Éléments de mathématique, not the Stacks project, and it is written by Nicolas Bourbaki, not Johan de Jong. – Harry Gindi Feb 10 '19 at 18:09
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    @YemonChoi Kevin Buzzard's email can be found here: http://math.columbia.edu/~dejong/wordpress/wp-content/uploads/2010/10/Kevin-Buzzard – Chromatic Homotopy Theory Feb 10 '19 at 18:10
  • @HarryGindi I meant it as a massively-collaboratively project, such as to not burden one specific person to write the whole thing. Also, it would still be useful to have (at some point) a tag system like the Stacks project (i.e. Gerby) since the scope of such a homotopy theory project would be gigantic – Chromatic Homotopy Theory Feb 10 '19 at 18:14
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    @ChromaticHomotopyTheory I don't mean to be harsh here, but for such a project to work, you need a leader with a vision. And I don't think you're leading anyone anywhere with an anonymous account on a (distinguished-as-it-is) Q&A website. – Harry Gindi Feb 10 '19 at 18:18
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    Even more than a vision you need a leader in charge. Someone who's willing to devote massive amounts of time to make sure that the project progresses, stays on track and keeps up the standards of quality. It also wouldn't hurt if it was someone with some prestige to begin with. This is no small thing to ask someone – Denis Nardin Feb 10 '19 at 18:20
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    @HarryGindi Referring to what you said in the chat room, I'm sorry if this question annoyed you. I shall mind you more when asking questions in the future. Regarding Denis Nardin's remarks, I can imagine how hard and time-demanding it is to write a textbook, that is why I suggested having a collaborative project, in which many people would contribute. – Chromatic Homotopy Theory Feb 10 '19 at 18:44
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    @ChromaticHomotopyTheory Unfortunately, even though a woman can make a child in nine months, nine women cannot make a child in a month. – Denis Nardin Feb 10 '19 at 18:57
  • That's a straw man argument. People can separately write parts of a project, with multiple persons coordinating it. For a silly example, to build a rocket, there are many things to do: get funding, design it, build the pieces, etc. All these parts are done by different people, but they are not coordinated by a single person, but by a team. – Chromatic Homotopy Theory Feb 10 '19 at 19:10
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    @ChromaticHomotopyTheory This is my last message here but: while I'm not an expert on how rockets are built, I find it extremely unlikely that there's no one ultimately in charge of the whole affair. All the tasks I mentioned in my first message require someone where the buck ultimately stops. What about deciding what the table of contents should be? Which foundations are best taken for this or that argument? If you hope to get a clear consensus on these things via a mailing list of many tens of authors... good luck. – Denis Nardin Feb 10 '19 at 19:15
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    For context: I am currently writing a paper with eight other coauthors. It's going excruciatingly slowly and not because anyone wants to slow the process (everyone has been great to work with). It's just that hard to make large-ish amounts of people work together. In fact I'd rather write such a project as you propose completely on my own than try to be the coordinator... – Denis Nardin Feb 10 '19 at 19:16
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    @DenisNardin I didn't know it was so difficult to collaborate with large numbers of people. Thank you for the real example. (I mean this with no irony whatsoever. Thank you, really) – Chromatic Homotopy Theory Feb 10 '19 at 19:28
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    @ChromaticHomotopyTheory People have fulltime jobs as CEOs, managers, and military officers for a reason. Organizing groups of people to all pull in the same direction is a fulltime job. – Harry Gindi Feb 10 '19 at 19:34
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    I've deleted two comments which are arguably argumentative, and if this continues might shunt the comments off to a chat room. – Todd Trimble Feb 10 '19 at 20:06
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    I think the closest you can get to equally subdividing the work is when one author writes all the statements and the other contributors fill in the proofs. Still, the former author would have a serious task on their hand. Splitting that task among several people is almost guaranteed to lead to inconsistent notations, missing/duplicate stuff and worse. – darij grinberg Feb 11 '19 at 05:56
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    (Great cases can be made for collaborative refining of more or less finished texts, however. For instance, Lang's Algebra would be a much better text if a dozen or so mathematicians who taught from it and filled the gaps with handouts would give it a proper edit.) – darij grinberg Feb 11 '19 at 06:01
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According to the about page of Kerodon:

Kerodon is an online textbook on categorical homotopy theory and related mathematics. It currently consists of a single chapter, but should grow (slowly) over time. It is modeled on the Stacks project, and is maintained by Jacob Lurie.

I think this may answer your question?

Will Sawin
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  • pupshaw said this in the comments at the same time... – Will Sawin Feb 10 '19 at 18:04
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    While I hope that kerodon will become a great resource, maybe it's not obvious to outsiders that the scope of kerodon is, in fact, quite restricted compared to what this question is asking. For example chromatic homotopy theory, unstable homotopy theory, and motivic and equivariant homotopy theory are all outside of kerodon's stated scope. And this is not even touching on whether you think algebraic K-theory is part of homotopy theory.... – Denis Nardin Feb 10 '19 at 18:11