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.
Best Answer
Easy. Who's gonna write it?
JDJ (Johan de Jong) has written almost the entire stacks project himself.