# Homotopy as a general organizing principle

One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories comprise very general ideas that arise in a variety of places. To quote Mike Shulman:

Personally, I don’t find it especially surprising that homotopy theory has more than one application to some other subject, any more than I would find it surprising that category theory does. I think it’s becoming increasingly clear that both of them are general organizing principles of mathematics.

I would like to build a list of interesting examples of this, especially unexpected applications of the ideas of homotopy theory in otherwise far removed areas of mathematics.

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Dan Isaksen once told me that "Model categories are for computation." I did not understand that for a long time, but it is a very insightful comment. I think that the cotangent complex is an excellent example of this philosophy, as well as what you are looking for. –  Sean Tilson Jul 16 '13 at 5:51
Are there applications to probability or analysis? –  Tom LaGatta Jul 16 '13 at 20:55
–  Alberto García-Raboso Jul 17 '13 at 2:52
Broad applicability seems like an over-statement here. What we are seeing is some applicability and some very young mathematics whose impact is yet to be seen. –  Ryan Budney Jul 17 '13 at 5:24
@Alberto: That link consists of 7 pages of definitions and 1 page with a (rather mechanical) proof. It is just a bunch of formalism, not an "application" to probability or analysis. –  user36938 Jul 17 '13 at 12:31

Algebraic K-theory would be a huge example. "Unexpected" is in the eye of the beholder, but I would imagine that every intrusion of homotopy theoretic ideas into the subject was unexpected by many.

An interesting classical example is the theory of "scissors-congruence" (http://ncatlab.org/nlab/show/scissors+congruence), which can be interpreted in terms of K-theory (Zakharevich, http://arxiv.org/abs/1101.3833).

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Charles, I am not sure I understand your answer. As (part of) algebraic K-theory started as part of homotopy theory in Whitehead's simple homotopy theory, homotopy theory's application there is not surprising. Also I think that Whitehead had an idea that Alg. K-theory had wide applicability. –  Tim Porter Jul 18 '13 at 6:31

Over on the nLab the organizing idea is that homotopy theory and higher category are what organizes mathematics. (As in What is.... the nLab?). As a deliberate pun on Wikipedia's "neutral point of view" this got called the "n point of view".

Here "higher category theory" and the $n$™ refers $(\infty,n)$-category theory hence in particular also homotopy theory ($(\infty,0)$-category theory).

For some reason, much of the discussion we had about it was about how to talk about it without upsetting those people who don't want to believe it... In any case, we started making some pages with lists of examples. such as

Each time I look back at these entries I realize how imperfect they are, in spite of some time invested into them. That's how it goes. But it's a start.

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I can't say anything about the historical context, but one of my favorite examples is topological fixed-point theory. The fixed-point index of a smooth endomap of a finite-dimensional manifold has a purely geometric description as a sum of the indices of all fixed points. But not only does it turn out to be homotopy-invariant, it turns out to be an instance of the general notion of symmetric monoidal trace, applied to suspension spectra in the stable homotopy category! Moreover, this perspective makes it much easier to prove abstract properties of the fixed-point index and its various generalizations.

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In my answers I tried to avoid applications of homotopy theory within topology, because I didn't think that counted as "far removed from homotopy theory." This answer is cool because it's an application which surprised topologists, though I'm not 100% sure a non-specialist would be surprised that homotopy could be used to detect the fixed-point index. In any event, it's good you came upon this thread since it started with a quote of yours. What other things did you have in mind when you made that quote? –  David White Jul 16 '13 at 20:57
I think the classical definition of the fixed-point index could reasonably be considered "geometry" rather than "topology". Furthermore, I think the classification of homotopy theory as part of topology is a historical relic --- homotopy theory is actually the theory of $\infty$-groupoids, and it's a historical accident that people first learned how to study $\infty$-groupoids by presenting them in terms of topological spaces. (One might even argue that it's an accident of set-theoretic foundations that this is even possible. (-: ) –  Mike Shulman Jul 17 '13 at 4:48
Would it be to early to argue, that the notion of homotopy (infinty groupoids) is unavoidable once the notion of identity has been properly formalised? And since identity is ubiquitous so will homotopy eventually be? –  Michael Bächtold Jul 17 '13 at 9:59
Well, it would be provocative. I'm not sure I would say exactly that, because there's a lot of wiggle room in the word "properly". –  Mike Shulman Jul 17 '13 at 16:53
I also find it amazing. I'm just saying I wouldn't assert that ML identity types are the "proper formalization of identity". –  Mike Shulman Jul 20 '13 at 0:30

A good place to start would be the following two places where a Fields Medal was awarded to a homotopy theorist:

1. Quillen's invention of Higher Algebraic K-theory and use of model category language to do resolutions in a context where they weren't possible before (which led to new computations, e.g. K-theory of finite fields). Quillen used model categories to prove that the derived category $D(R)$ is triangulated, a fact which had not been known previously. It also allowed him to use homotopy theoretic methods for a much wider class of rings.
2. Motivic Homotopy Theory to solve the Milnor Conjecture and to reframe certain other famous number theory conjectures. This has the added benefit of allowing one to "do homotopy theory" with schemes.

Many nice references if you want to read more about the latter can be found at this MO question.

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What do you mean by "Quillen used model categories to prove that $D(R)$ is triangulated, a fact which had not been known previously"? The État 0 of Verdier's thesis is from 1963. Are you alluding to set theoretic difficulties with the unbounded case? –  Martin Jul 15 '13 at 15:29
@Martin: yes, precisely. To get around the set-theoretic issues you need a calculus of fractions, which is just what the model structure gives you. I suppose my notation was a bit unclear. If I'm talking about the bounded derived category I usually write $D^b(R)$ or some such thing. When I write $D(R)$ I generally mean unbounded. Perhaps algebraists have a different notation –  David White Jul 15 '13 at 15:33
Thanks for clarifying -- I think the notation is fine (I usually add "unbounded" to avoid ambiguity). I guess I was slightly thrown off balance by what I read to be an emphasis on the triangulated structure rather than the size of the Hom-sets. –  Martin Jul 15 '13 at 15:48

Having a homotopy theory for Operads gives a very general framework to talk about rectification, where you beef up the amount of algebraic structure your object has. In particular, once you realize $A_\infty$ as the cofibrant replacement of $Ass$ and once you put model structures on $A_\infty$-alg and $Ass$-alg (i.e. the full subcategory of associative algebras), then rectification is just the existence of a left Quillen functor. Similar things hold for $E_\infty$-alg and $Comm$-alg (again, if both admit a model structure) and for $L_\infty$-alg and $Lie$-alg. A good place to read about this is Berger-Moerdijk Axiomatic Homotopy for Operads (2003). The same authors later worked out the $W$-construction (or Boardman-Vogt resolution) in much more generality, which allows one to construct cofibrant replacements of operads by hand, rather than relying on general existence theorems.

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How is this an application of homotopy theory to an "otherwise far removed" area of mathematics? I would have thought the idea of rectification is already homotopy-theoretic. –  Mike Shulman Jul 16 '13 at 17:44
Well, the ideas of $A_\infty$ and $E_\infty$ algebras came up naturally for people doing Rings and Algebras, and I think this was independent of the operadic language. There was a whole series of conferences on this sort of thing over the past year, culminating with the Recent Trends in Rings and Algebras conference in Murcia last month. I can't comment too much on the history, but I've talked to a number of people who care about $A_\infty$ algebras but don't know what a model category is (though have heard of them because of these surprising applications) –  David White Jul 16 '13 at 19:22
I don't think you have to know what a model category is to be doing homotopy theory. By that measure, no one was doing homotopy theory until 1967. (-: And you certainly don't need to know what an operad is. –  Mike Shulman Jul 17 '13 at 5:14
I guess my point was: they don't think they're doing homotopy theory. These things came up naturally for them in the context of doing pure algebra. –  David White Jul 17 '13 at 12:16
I agree that applications of $A_\infty$ and $E_\infty$ notions in algebra are an application of homotopy theory to a different area of mathematics. But it sounded to me as though this answer was talking specifically about the application of operadic ideas. I was just saying that the application of homotopy theory is in the initial realization that an idea like homotopy-coherence or rectification is important, not in any later utilization of some particular homotopy-theoretic technique like operads. –  Mike Shulman Jul 17 '13 at 16:58

Jacob Lurie just finished teaching a course at Stanford titled Tamagawa Numbers via Nonabelian Poincare Duality. You can read the definition of a Tamagawa number here (I know I'd never heard of them before). Based on that, I'm pretty surprised homotopy theory (in the guise of quasi-categories now, not model categories) comes into play here, but again it's as a unifying and organizing viewpoint which allows you to construct things purely formally via universal properties. This way of thinking has got a lot of people hopeful that Lurie's machinery can be used on all sorts of algebraic problems.

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This "answer" entirely misses the real content. The substance of Lurie's arguments is not in formal constructions via universal properties. He brought in deep insight from topology and serious ideas in algebraic geometry. Modulo issues of parahoric models of simply connected groups, the Tamagawa volume in the function field case is a point-count on a certain gigantic stack over a finite field, so in view of the Grothendieck-Lefschetz trace formula one should expect that cohomology on those stacks is relevant. Lurie uses homotopical ideas to deeply understand the structure of such cohomology. –  user36938 Jul 16 '13 at 5:56
It's community wiki. Please feel free to edit my answer and insert your insights. I certainly don't claim to be any kind of expert on Lurie's work. I'd love to learn more –  David White Jul 16 '13 at 11:18
The way to learn more is not by posting a content-free "answer" and waiting for other people to edit in some substance; it is by doing the hard work of reading Lurie's very instructive lecture notes for that course. Moreover, the comment about "proving the Geometric Langlands program" is pure hype. –  user36938 Jul 17 '13 at 12:01
This is getting too vitriolic for me. Stopping to care...now –  David White Jul 17 '13 at 12:16
@user36938 I don't think David is averse to doing hard work, but part of the point of MO is to share such insights that make the learning process easier. I think he is merely inviting you or others to share them -- if you want. (It's fine to criticize an answer of course, but let's try to keep the criticism as constructive as possible. The words before the semicolon in your last comment could be read as a put-down, or as verging on rudeness.) –  Todd Trimble Nov 28 '13 at 22:31

Finnur Lárusson has developed holomorphic homotopy theory in his work on Oka Theory. I really know nothing about it, but the details can be found here.

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Where's the beef? Does this stuff tell us anything about complex manifolds or complex-analytic spaces that wasn't known long ago? –  user36938 Jul 17 '13 at 12:16

Cardinal arithmetics is usually considered far removed from homotopy theory, so perhaps this would be an example.
Several cardinal invariants in set theory may be viewed as derived functors, in a very degenerate model category setting. is usually considered remote from homotopy theory. See Gavrilovich, Hasson, A homotopy theory for set theory, Part I, Part II.

In fact, there is also the following which appears to be an explicit attempt to that the ideas of homotopy theory have very broad applicability in mathematics. Gromov, In search for a structure. Part 1, On Entropy", talks entropy in terms of category theory, but perhaps not necessarily homotopy theory. He has the following to say in a postscriptum: Apology to the Reader. Originally, Part 1 of ”Structures” was planned as about a half of an introduction to the main body of the text of my talk at the European Congress of Mathematics in Krak´w with the sole purpose to motivate o what would follow on ”mathematics in biology”. But it took me several months, instead of expected few days, to express apparently well understood simple things in an appropriately simple manner. Yet, I hope that I managed to convey the message: the mathematical language developed by the end of the 20th century by far exceeds in its expressive power anything, even imaginable, say, before 1960. Any meaningful idea coming from science can be fully developed in this language. Well..., actually, I planned to give examples where a new language was needed, and to suggest some possibilities. It would take me, I naively believed, a couple of months but the experience with writing this ”introduction” suggested a time coeﬃcient of order 30. I decided to postpone.

There was also talks by Voevodsky (and older work by Tsvetkov(?)) who consider a category theory approach to probability. (There is an online video talk by Voevodsky on this in Russian. I'll try to recall the name of the book by Tsvetkon(??))

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Interesting! Do you know of any written references for the talks of Voevodsky and Tsvetkov you mention? –  Michael Bächtold Jul 17 '13 at 9:56

Another place where model categories have been used in algebra is in the study of Hopf-Galois extensions, which pop up naturally in many places, including algebraic geometry, noncommutative geometry, and Galois theory over a commutative ring rather than a field. I don't know a ton about this story, but Kathryn Hess has been working on it for years. She (with Shipley) recently developed a homotopy theory of coalgebras over a comonad precisely to apply to this type of situation. Just like with Quillen, one of the things you really want is resolutions, and to make those exist with good functorial properties you need a model category

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