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I have deleted an old out of date query box from homotopy theory.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 -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 -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 -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 -action is, it turns out (hence “happens to be”) the A-type action on the 4-sphere, namely the one induced by regarding and identifying .
Moreover, analysis shows that what makes the gauge enhancement work is the fact that the fixed point set of this action is . But this is the case for every non-trivial subgroup of . Restricting to finite subgroups, this yields the ADE-orbifolds of the 4-sphere.
The power of !
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 -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/-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)
added pointer to today’s
added pointer to:
added pointer to:
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(I see that the ordering of the list of references is suboptimal, but I’ll leave it as is for the moment)
added pointer to:
I’ve added three references after Whitehead’s book: Mosher-Tangora; Aguilar-Gitler-Prieto; and Strom. Each of these (and Whitehead’s book) treats the homotopy theory of topological spaces, so I’ve changed the heading for these references to “Textbook accounts of homotopy theory for topological spaces”.
I added these because I think the current description of homotopy theory on this page is incomplete, but I don’t know how to complete it. As I understand it, another definition of homotopy theory would be the collection of tools developed to study the homotopy groups of spheres. Then I would say that the theory of -categories is an extension of some of those tools to more general contexts. But things like the Steenrod Algebra, Bott Periodicity, and the LHS spectral sequence are part of what many would consider the central subject of homotopy theory. I know that one can extend some of these more sophisticated constructions to other model categories or -categories.
As I said above, I don’t have good ideas for how to rewrite the article with a more balanced point of view. I think it’s a difficult task, and I’m not suggesting that someone else should do it for me. Instead, I wanted to add these three references for people (e.g. students) who will be looking for further explanation of homotopy theory in the sense I (and I think many others) understand it.
It might also make sense to add these to the page called “homotopy theory of topological spaces”, but I wanted to stick to one edit at a time since it’s my first! And I hope you can see why I would disagree with putting these references only on the topological spaces page.
Niles Johnson
There is some overlap with the references at algebraic topology and elsewhere. But by all means, please add references, here and elsewhere, thanks.
I have made your author hyperlinks to Robert Mosher and Martin Tangora come out now.
I have always been citing Gitler et al’s “from a homotopical viewpoint” without hyperlinking his coauthors. Forget why, I guess I didn’t find homepages for them to link to. But now that you request these pages, I’ll add them.
further on the References-section here:
have re-organized slightly, to have all the point-set homotopy textbooks in one block;
more ought to to be done further below that block, but I am out of energy now
but added the following line at the beginning:
(See also the references at algebraic topology at simplicial homotopy theory and at -category theory.)
and have added similar lines to the References-sections of these entries
If anyone is ambitious, we should have an entry topological homotopy theory to go alongside simplicial homotopy theory. That would be the place to add further pointers also to equivariant and global homotopy theory.
To explain, this edit #24 arose from a Twitter discussion where some suggested that the nLab take on homotopy theory is one-sided.
Daniel Kehlman captured this neatly in his essay collection “Lob. Über Literatur”: After checking up on the user profiles of online critics, he concluded:
Erfolg bedeutet, dass auch die Zwetschge Sumsi eine Meinung über sie hat.
Hi Niles, where on Twitter is it that you are chatting? The link in #29 gives me just a thread with 4 tweets, none of which by you.
I have found further messages now, but still nothing from you. But I’ll stop searching now.
Sorry, I gave the root of the tree. Niles speaks here.
Thanks. I had missed an intermediate branch. Twitter has suboptimal threading support.
But seeing this, I now get the impression that the issue you felt like raising was not really with the References-section, but rather with the Idea-section: You are trying to say that you don’t usually think of homotopy theory as abstract-, but as topological-homotopy theory. Is that right?
In any case, I have touched the Idea-section:
added some sub-sections to delineate the different aspects developed there,
added some graphics
added a new paragraph that briefly explains how it is that people tend to use topological spaces to speak of abstract homotopy types (namely: due to cohesive HTT at work in the background). See here.
Oh, I think the distinction between “topological homotopy theory” and “abstract homotopy theory” is really helpful. You are right about what I was trying to raise, but not knowing how to articulate. Thanks for taking the time to understand what I mean. Maybe you’ve already seen them, but I think these remarks on twitter by John Baez are consistent with my view:
“…while it’s catching on in some circles, I think the idea that “homotopy theory is not a branch of topology” would shock most mathematicians.” here
“…I’m talking about mathematicians, topologists and even homotopy theorists on average [not about specific people on the Twitter thread]….” here
So I (and I think many people) would expect the unqualified “homotopy theory” to include those things described under topological homotopy theory, but also going much further into things like Toda brackets, Massey products, characteristic classes, and so on. Those kinds of things can (at least in principle) be developed in more abstract homotopy theory, but the constructions and applications are not as thoroughly developed.
Either way, I think the current version of the article is better for all readers than was the previous version :)
Okay, glad it helped.
I am not sure where you are headed with the “consistent with” somebody else’s Twitter comments. If this is a request for further edits, please say it more directly.
In any case, I have added one more paragraph to “General idea” (here), mentioning homotopy theory as embodying the “gauge principle in mathematics”.
No, not trying to request further edits. I was just giving some evidence that the perspective I was explaining is shared by others, to justify the edits you’ve already made. Thanks again for looking at it.
Just to say that it is commonplace that people like to conflate topology and homotopy theory, no need to call further witnesses. :-)
Ha ha! Fair enough :)
have now removed the previous content of the section “References” (here) and replaced it by !include-ing homotopy theory and algebraic topology – references.
So to further edit the list of references here, you need to edit there.
This question is also addressed in the homotopy theory FAQ: https://ncatlab.org/nlab/revision/homotopy+theory+FAQ#intro
Thanks for the pointer Dmitri! It seems that reasonable people can disagree on how broad the term “homotopy theory” ought to be understood, and that’s fine with me. I don’t really object to the description given in the FAQ, even if I view the subject differently.
I believe the initiator of that Twitter thread had some point to make about nLab’s presentation of homotopy theory other than whether and how it extends beyond the topological. If interested, there’s an 11 point thread beginning here. If misapprehensions of the subject are being derived by learners, as he claims, it’s not clear to me that it’s for us to address. If the concern is that we emphasise high-level conceptions over the messy calculations of the field, then this applies just about everywhere. A high school student won’t get the sense from integral that they have to know and apply a range of techniques to answer specific problems, such as .
As far as I can see there is no tangible content in the rambling tweets by this user and nothing in them specifically concerns the nLab, besides its name being dropped continuously. If and when any concrete comment shows up, I may try to react to it, but otherwise I suggest to not waste energy on kids who project their introspection onto the Twitter screen.
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