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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    $\begingroup$ 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: ncatlab.org/nlab/show/Handbook+of+Homotopy+Theory $\endgroup$ – Shay Ben Moshe Feb 10 '19 at 18:25
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    $\begingroup$ @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. $\endgroup$ – Najib Idrissi Feb 10 '19 at 18:42
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    $\begingroup$ @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! $\endgroup$ – Shay Ben Moshe Feb 10 '19 at 21:41
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    $\begingroup$ 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). $\endgroup$ – Timothy Chow Feb 11 '19 at 0:31
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    $\begingroup$ 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. $\endgroup$ – David Roberts Feb 11 '19 at 5:54

Easy. Who's gonna write it?

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

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    $\begingroup$ @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. $\endgroup$ – Harry Gindi Feb 10 '19 at 18:04
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    $\begingroup$ @YemonChoi Kevin Buzzard's email can be found here: math.columbia.edu/~dejong/wordpress/wp-content/uploads/2010/10/… $\endgroup$ – Chromatic Homotopy Theory Feb 10 '19 at 18:10
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    $\begingroup$ @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. $\endgroup$ – Harry Gindi Feb 10 '19 at 18:18
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    $\begingroup$ 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 $\endgroup$ – Denis Nardin Feb 10 '19 at 18:20
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    $\begingroup$ @ChromaticHomotopyTheory Unfortunately, even though a woman can make a child in nine months, nine women cannot make a child in a month. $\endgroup$ – Denis Nardin Feb 10 '19 at 18:57

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?

  • $\begingroup$ pupshaw said this in the comments at the same time... $\endgroup$ – Will Sawin Feb 10 '19 at 18:04
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    $\begingroup$ 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.... $\endgroup$ – Denis Nardin Feb 10 '19 at 18:11

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