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I have deleted an old out of date query box from homotopy theory.
have added a pointer to the new Problems in homotopy theory Wiki
I have expanded the beginning of the list of References at homotopy theory, with brief comments (Quillen had been missing, Lurie had been missing, UF had been missing).
The Idea-section at homotopy theory had been abysmal.
I have written something more substantial now, see here.
This is just so that I may point from other introductory pages like Introduction to Topology to “homotopy theory” without feeling that I am sending the reader down the cliff. But I don’t actually have time to deal with bringing the entry on homotopy theory into shape right now. Everybody please feel invited to expand on that entry further.
below the reference to Tyler Lawson’s “Homotopy theory: The future” I added pointer to Clark Barwick’s “The future of homotopy theory” (now Barwick 17).
It seems to fit there, even though it is less about the subject of homotopy theory than about the sociology of its practitioners.
It would be helpful if Barwick could give some existing exemplary writings of the kind he’s looking to encourage.
Presumably Lawson’s talk would count. Strictly, I guess it should be named ’Chromatic homotopy theory: The future’, since it came at the end of the 2013 Talbot workshop of that name.
It’s time to finally nail down the fundamental relationship between homotopy theory and string/M theory from first principles, not relying on oracles.
With Hisham and John H., we have finally sorted out how ADE-equivariant homotopy theory classifies intersecting black brane configurations. (It’s closely related to, but not exactly how I had imagined it in Jan 2016, instead it’s in un-stable equivariant (rational) cohomotopy, after all…).
First we were perplexed that the $G_{ADE}$-equivariant enhancement of the M2/M5-cocycle, while correctly detecting the “black” M2 and M5, misses the KK6 (the “M6”) as well as the M9. But reflection reveals that the nascent equivariant homotopy theory of black branes is already smarter than we are: The isolated KK6 (in its guise as the D6) suffers from the RR-field tadpole anomaly and it is only its “M5 half-brane“-intersection with the M9 (it its guise as the O8-plane) that should appear, which turns out to be exactly the thing that the $G_{ADE}$-equivariant cohomology theory reports as being a non-trivial cocycle.
How does one know about ADE gauge enhancement from “first principles”? I see from your slide 54 that one construction “happens to be the same as” a black M5-brane at an A-type singularity, but that doesn’t sound like a first principles derivation.
You are referring to slide 54 here.
So there is one choice of $S^1$-action on the 4-sphere such that gauge enhancement exists in the sense of a lift as on slide 61, after fiberwise Goodwillie linearization. This is from “first principles” in that it is just a mathematical analysis of the structure inside the equivariant Whitehead tower that grows out of the super-point.
That $S^1$-action is, it turns out (hence “happens to be”) the A-type action on the 4-sphere, namely the one induced by regarding $S^1 = U(1) \subset SU(2) = S(\mathbb{H})$ and identifying $S^4 =S(\mathbb{R} \oplus \mathbb{H})$.
Moreover, analysis shows that what makes the gauge enhancement work is the fact that the fixed point set of this action is $S^0 \hookrightarrow S^4$. But this is the case for every non-trivial subgroup of $SU(2)$. Restricting to finite subgroups, this yields the ADE-orbifolds of the 4-sphere.
The power of $\emptyset$!
I suppose now you are referring to slide 83.
Any further thoughts on expressing such work (#7) in terms of “Cartan geometry with singularities” from here?
I haven’t further developed “Cartan geometry with singularities” yet, but our understanding of these local models (super-Minkowski spacetimes with super-ADE-singularities) has much progressed. We should have a first version of the article to share next week.
It turns out that the classification of these local models matches the classification of BPS-solutions to 11d supergravity, including a fair bit of fine print. This means that when working in supergeometry, the “quantum numbers” of black brane solutions to supergravity (dimension, BPS degree, singularity structure) is already fixed by a super tangent space wise analysis, hence on the local model spaces.
This reinforces the idea that black $p$-brane physics is a topic in super Cartan geometry with singularities. On a rough level it is clear how this works, but I haven’t tried to write it out as a formal theory yet.
When writing cross-discipline, one notices how few background texts there are that communicate cross-discipline. In finalizing our Equivariant homotopy and super M-branes for publication, I need for the very first paragraphs one-punch-knockout citations: “string/M-theory is [cite and done], homotopy theory is [cite and done]” to quickly orient the reader who may still hop off and not make it to the more detailed reviews provided down the line in the text. For the string/M-theory bit I am fairly happy with pointing to Duff 99.
But for the homotopy theory bit?
Most “introductions” to homotopy theory/$\infty$-category theory jump deep into technical details of simplicial sets and the like right away. That’s not what the expected mathematically inclined string theorist wants or even needs to see, who hasn’t even been told yet what “homotopy theory” as such actually is.
I tended to like to cite Mike’s “The Logic of Space” for the purpose of broader introduction to what it’s all about. But while very nice in itself, it seems too much focused on type theory for my intended audience.
What to do? Any text out there that gives a modern idea of “homotopy theory” to the educated layman who is mathematically inclined but really can’t be expected to enjoy hearing things like “the cofibrations are simply the presheaf monomorphisms” before he has even been told what homotopy theory as such actually is?
The problem may be what prerequisites should be assumed. I looked back at a paper I wrote on Abstract Homotopy Theory for the Chilean journal Cubo, but it assumed a basic knowledge of algebraic topology. My book with Kamps also has something like the same set of prerequisites. Ronnie’s book Topology and Groupoids starts from much further back but then gets nowhere near the more infinity categorical stuff, and so on. If one assumes too much, the basic idea risks being submerged and if not enough, then the development will take too long. I am mentioning these books and papers since I know them well, not to encourage sales!
I tried to do something with the Menagerie Notes but those take quite some time to get where you would want them to be and also ignore stable homotopy theory as I am not a great fan of that area!
I can offer the tex files for the Menagerie as a base from which to build using ’copy-paste-edit’ if that would help, also perhaps the n-Lab could use what you, Urs, have already put online together with other material to produce a pdf file or a Kerodon type intro that would do what you need.
Any text out there that gives a modern idea of “homotopy theory” to the educated layman…
Sounds like one of the things Clark Barwise was calling for in The future of homotopy theory (pdf) [hmm, seems to have just disappeared from his site].
I wonder if the Handbook of Homotopy Theory will provide such a thing. The old Handbook of Algebraic Topology is perhaps seeing it too much as part of alg top, as the title suggests.
added this pointer:
added publication data for these items:
Emily Riehl, Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)
Birgit Richter, From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (doi:10.1017/9781108855891, book webpage, pdf)
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