How to explain homotopy theory to a number theorist

(Although I tried to make this an impersonal account of an ideal-typical state of affairs, this will actually be semi-autobiographical.  Readers are asked to pretend not to notice.)

I missed my chance to learn homotopy theory as a student but on three occasions made half-hearted attempts to make up the gap in my understanding.  The first was in connection with the Beilinson conjectures on special values of L-functions, which I tried to understand in preparation for my stay in Moscow during the academic year 1989-90.  I had the pleasure of reading, and more or less understanding, Quillen’s magnificent paper Higher algebraic K-theory: I, but when I arrived in Moscow I discovered that Beilinson was spending most of his on the road and my Russian colleagues had mainly moved on to other things (especially quantum groups).  I first met Vladimir Voevodsky in a corridor of the Moscow State University building (where he was an unofficial student) where I was sitting in a corner trying to learn about spectra.  Voevodsky went on to do fundamental work on motivic homotopy theory in connection with his construction of a (triangulated) category of mixed motives.  I, on the other hand, soon realized that the foundational material in homotopy theory played essentially no role in work on Beilinson’s conjectures after Beilinson’s original paper, and stopped thinking about homotopy altogether for 17 years.

In 2007 I learned that some homotopy theorists were studying my book with Richard Taylor on the local Langlands correspondence in the hope of finding information relevant to the computation of homotopy groups of spheres (and the like).  There’s actually an interesting mathematical background to this story.  In the early 1990s, (homotopy theorist) Mike Hopkins and (number theorist) Dick Gross wrote a paper, motivated by considerations in homotopy theory, that has had considerable influence in number theory.  My book with Taylor made heavy use of the theory developed in the book by Rapoport-Zink, which in turn was strongly influenced by the Hopkins-Gross paper.  After looking into the question, I determined that my specific expertise would be of no use to the homotopy theorists and I stopped thinking about this as well.

A few years later, probably in 2009 or 2010, my third descent into homotopy theory began when Ed Frenkel explained to me in Paris that he and Dennis Gaitsgory had come to the conclusion that the local theory of the geometric Langlands correspondence should be formulated in terms of actions of groups on categories rather than on vector spaces.  There is a precise statement of this principle on p. 310 of his book Langlands Correspondence for Loop Groups, published in 2007.  Regular readers of this blog will recognize this as an early version of the top line of the blackboard reproduced in this earlier post.


Frenkel mentioned the principle to me in response to a question about formulating a correct version of the Langlands correspondence for p-adic groups modulo p (a relatively new branch of number theory that grew out of the work of Wiles and Taylor on Fermat’s Last Theorem and related questions).  I have nothing to say about how this principle applies to loop groups, but when I heard Gaitsgory’s talk at the Laumon birthday conference in 2012, I began to entertain the (possibly irrational) belief that the mod p local Langlands correspondence looks more like the middle line of that blackboard (the one that only involves (∞,1)-categories, not (∞,2)-categories).

This is the background to the question posed in the title of this post, and to my superficial familiarity with some of the terminology used by homotopy type theorists.  Only a handful of homotopy theorists have had any success at all in getting the point across to me, and the lessons have always been too short.  I think the only solution will be on the order of machine learning — I will keep attempting to cross the (virtual) room of homotopy theory and after a few thousand trials I hope I will be able to get to the other side without major damage.


9 thoughts on “How to explain homotopy theory to a number theorist

    1. mathematicswithoutapologies Post author

      Thanks for the corrected link. I’ve never met you but I tried to read one of your articles in connection with my second dip into homotopy theory, and got far enough to understand that my specialized knowledge would not help advance the chromatic program — if indeed there is such a program (this in response to Mike Shulman’s comment about the “big name” mathematicians who “set out ‘research programmes’ that everyone else is expected to follow”).

      Voynich is right about the nLab article on rigid analysis, but I suspect it doesn’t go far enough to help you with applications to homotopy theory. I am presently working on a different problem: how to explain perfectoid spaces, and in particular rigid analytic spaces, to an audience of historians and philosophers? And the question that should inevitably follow: what can the historians and philosophers say about contemporary mathematics if they don’t understand where perfectoid spaces fit in?


      1. David Roberts

        Well, I would start with p-adic numbers, how they are sort-of-but-not-really like real numbers, and then discuss the nastiness that is analytic geometry (e.g. p-adic discs) without the perfectoid approach. For analogy, compare Weil’s dream of treating varieties over finite fields with geometric means: needed a rethink of what geometry means, and we got schemes.

        But I’m sure you know this, but are also trying to not do a David “of course I can explain schemes to biologists” Mumford.


      2. B3

        I suspect that it might be more useful for homotopy theorists to know what rigid geometry is good for (instead of understanding the basic definitions of rigid geometry). For instance, homotopy theorists seem sold on formal schemes, so an exposition outlining why rigid spaces are like “formal schemes up to isogeny” could be worthwhile. Unfortunately, I do not know any such treatments…


      3. John Sidles

        From David Mumford’s own account Can one explain schemes to biologists?

        David Mumford laments  “The gap between the world I have lived in and that even of scientists has never seemed larger. […] The real problem is that such a huge and painful gap has opened up between mathematicians and the rest of the world.”

        An engineering view  David Mumford’s “huge and painful gap” originates in the global STEAM community’s too-prevalent denial and/or rejection of the huge and painful mission that modern mathematics has ordained: the maximal substitution of considerations of universality and naturality for considerations of physicality. And yet this mission can’t be evaded, or even long-postponed, because it’s already irretrievably underway.

        “Young 21st century mathematicians and scientists (and even young engineers) have nothing to lose — and a vast universe to gain in professional and creative opportunities —by systematically transcribing their old-fashioned [let’s face up to it!] 20th century physical understanding of quantum statistical mechanics and thermodynamics into lively 21st century mathematical terms of universality and naturality.”

        Conclusion This huge-yet-painful, transgressively transformative, and just-beginning 21st century STEAM enterprise — the transcription of physicality to universality and naturality — is motivating more-and-more engineers (me for one) to read the mathematical literature (including MWA) with deliberately deconstructive scrupulosity.


    2. Charles Rezk

      I think B3’s comment is right on target: why rigid analytic spaces are “formal schemes up to isogeny” is (probably) exactly what I’d like to know.

      Also: there seems to be a superficial analogy “formal schemes : rigid analytic spaces” vs. “schemes over C : complex analytic spaces”. Is there anything more to this, or is it a purely facile relation? (I can never tell when I should take such analogies seriously.)

      Operads, on the other hand, are easy! They are just a formalism for navigating through a certain part of the parameter space of possible algebraic structures.


  1. Jack Morava

    Dear Michael,

    Re your (admirable) efforts to come to terms with the interactions between number and homotopy theory: I think a lot about my old drinking buddy Will Blake, who said (in,
    The Marriage of Heaven and Hell )

    “If the fool would persist in his folly he would become wise.”

    I don’t mean this as snarkily as it may sound: I think of it
    mostly in reference to myself. I tried to use it as the epigraph
    to a section of my

    Noetherian localisations of categories of cobordism comodules.
    Ann. of Math. (2) 121 (1985) 1–39

    but was made (probably wisely) by the referees to take it out…


    (:+{)} Jack



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