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.