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Of course, personally, I think the answer is a big Yes!

However once, a while ago, while giving a talk about higher category theory, I was asked a question about whether higher category theory was useful outside of the realm of higher category theory itself. I was asked if there was anything that can be proven using higher category theory that couldn't be proven without it?

I think it is a somewhat common to experience this sort of resistance to higher categories, and I think this is a fair question, at least if you also allow for insights gleaned from a higher categorical perspective that would not have been possible or obvious otherwise.

I have a small handful of answers for this question, but I certainly don't feel like I know most of the applications, nor the best. I thought it could be useful to compile a big list of applications of higher category theory to other disciplines of mathematics.

Question: What are useful applications of higher categories outside the realm of higher category theory itself? Are there any results where higher categories or the higher categorical perspectives play an essential role?

Here I want "higher category" to be interpreted liberally, including various notions of n-category or $(\infty,n)$-category. I am not picky.

I also want to interpret "essential" to just mean that it would be hard to imagine getting the results or insights without the use of higher categories, not in some precise mathematical sense. But, for example, saying "homotopy theory is just an example of the theory of $(\infty,1)$-categories" doesn't really count.

The usual big-list rules apply: This is community wiki, and please just one application per answer.

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Here's a thought: [Higher category theory : Mathematics] = [Mathematics : Science]. I was inspired by xkcd.com/435. More seriously, is it maybe the case that the reactions of generic mathematicians towards higher category theory are similar to those of generic scientists towards mathematics?... –  André Henriques Dec 8 '11 at 19:54
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"Are there any results where higher categories or the higher categorical perspectives play an essential role?" Higher categories play an essential role in making me thrill. (Although I must admit that this result may be reached by other means.) –  Jonathan Chiche Dec 8 '11 at 20:52
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André, your comparison is very very skewed: it is a safe bet that most of mathematics has absolutely no use whatsoever of higher categories! –  Mariano Suárez-Alvarez Dec 8 '11 at 21:46
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@Buschi and @Andre : I think that statements like that are part of the reason that some mathematicians are a bit skeptical about higher category theory. –  Andy Putman Dec 9 '11 at 4:10
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@Dmitri : I think that the onus of providing evidence is on someone making an extraordinary claim (eg that higher category theory is useful in most of mathematics). That being said, it is easy to enumerate huge swaths of math in which higher category theory has hitherto played no role. For example, analysis, Riemannian geometry, geometric topology (other than quantum topology, which is really a different subject), dynamics, geometric group theory, analytic number theory, probability theory, etc. And even in areas where it plays some role (eg algebraic geometry), most people don't use it. –  Andy Putman Dec 9 '11 at 16:30
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7 Answers

I am very much used to these kind of questions. Are 2-categories useful? What can one prove using gerbes? Why should I care about stacks?

I think a funny way to react to these kind of questions, with often surprising results, is to return the question:

If you want to know what higher X is good for, explain first what X is good for, in your opinion.

And whatever the person answers, I found it mostly very easy to generalize the given argument from X to higher X.

Example 1 If X is "category", a common answer is "it keeps track of the automorphisms of the objects". Well, a 2-category keeps track of the automorphisms of automorphisms.

Example 2 The question was: "What can you prove with gerbes?", so I'll reply: "What can you prove with bundles?". People are often completely puzzled by this question, so they'll accept that a notion may be useful even if it's not there to prove something.

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I like very much this answer. –  DamienC Dec 8 '11 at 21:32
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Yes, very socratic. –  Joël Dec 9 '11 at 3:38
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You're not really answering the main question: "are higher categories useful?" You're just giving ways of intimidating skeptics... which is useful ;-) –  André Henriques Dec 9 '11 at 10:49
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Andre: no, I think he's saying that in order to answer the question, first you find out what sort of answer the asker is looking for, and then you can give it to them. –  Mike Shulman Dec 9 '11 at 21:58
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A great example of the usefulness/necessity of bicategories is the theory of parametrized duality in May-Sigurdsson's book Parametrized homotopy theory, and the associated notion of trace which Kate Ponto and I have been working on.

Spanier-Whitehead duality can be described in purely point-set topological language, but it becomes much clearer when phrased as duality in the symmetric monoidal stable homotopy category. The Lefschetz fixed-point theorem then becomes simply a consequence of the functoriality of homology.

The most useful parametrized version of Spanier-Whitehead duality, called Costenoble-Waner duality, can also be described in purely topological language, but as May and Sigurdsson realized, it becomes much clearer when phrased using adjunctions in a bicategory of parametrized spectra. In particular, certain dualities which are quite tricky to construct explicitly now follow from purely formal considerations. More refined fixed-point theorems involving the Reidemeister trace and the Nielsen number also follow from formal considerations, cf. here.

I realize that bicategories are a bit low-dimensional, as higher-dimensional categories go, but there are some indications in May-Sigurdsson of a need to go at least one level up to some sort of tricategorical structure. And of course all of this is happening homotopically, so really we are at $(\infty,2)$-categories or $(\infty,3)$-categories.

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The notion of Morita equivalence (in its various incarnations: algebras, $C^*$-algebras, von Neumann algebras, Poisson manifolds, Lie groupoids, orbifolds, algebraic stacks) is illuminated by higher category theory. Ok... not so high. See e.g. this paper for a short survey.

That's an example of a notion that was first formulated without reference to higher categories, and later explained using that language.

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I completely agree that bicategories is the right language for Morita theory, but just to play the devil's advocate, is it essential like the questions asks? –  Benjamin Steinberg Dec 8 '11 at 20:05
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It does help explain why there are so many different ways to 'invert Morita equivalences', because all the known constructions (and in fact any possible construction) is an example of a localisation of a bicategory. The higher category theory here is essential - one can't even state the required universal property without bicategories, and implicitly, the tricategory of bicategories (need the internal homs). –  David Roberts Dec 8 '11 at 20:28
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@Dave, thanks. I'll use this next time I have to justify why bicategories is the right language for Morita equivalence to a ring theorist. –  Benjamin Steinberg Dec 8 '11 at 20:42
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This is a great answer which shows that higher category enables us to unify various ideas in mathematics. But I think that the question is about essential insights which relied on this higher perspective but can be formulated without higher categories. Any examples here? –  Martin Brandenburg Dec 9 '11 at 8:14
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The point isn't that anything "relies on higher category theory" but rather that people were already doing higher category theory in the first place. You can always formulate something without using a particular word if you're willing to do enough work. –  Noah Snyder Dec 12 '11 at 0:52
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If you are a low-dimensional topologist and you care about skein modules (e.g. those which are related to generalized Jones polynomials), then you should also care about higher categories. To compute the skein module of a manifold that has been cut into several pieces you need to use higher categories in an essential way.

Very similarly (and much less surprisingly), if you are interested in computing low codimension TQFT invariants (vector spaces and linear maps), then you should also care about the high codimension TQFT invariants (n-vector spaces), as these are often the most efficient way to do computations.

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This remembers me of Lurie's remarks in his TQFT paper: In order to keep track of all the lower-dimensional pieces a higher-dimensional manifold is made up out of, we really should replace the 1-category of bordisms by a higher category. –  Martin Brandenburg Dec 9 '11 at 8:23
    
Do you have a reference for the statement in your first paragraph? –  Peter Samuelson Dec 12 '11 at 1:01
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I have been doing what I consider to be higher category theory since the 1970s. It went under the name of homotopy coherence theory and so was thought of as homotopy theory. I used Kan complexes etc.Which is it? The distinction is TOTALLY ARTIFICIAL so, Chris, I do rather object to setting up strange barriers between different areas of pure mathematics as this question requires.

That was more a comment, here is an answer:

Ronnie Brown and Phil Higgins' work on higher dimensional groupoids is a natural continuation of J. H. C. Whitehead's Combinatorial Homotopy II. The first of the two papers is where CW-complexes are introduced, so is that algebraic topology. Could RB and PJH have proved their higher van Kampen theorem without using higher dimensional groupoids? Possibly but they didnot!!! They used intuitions and methods from higher dimensional category theory .... including ideas from Grothendieck, Ehresmann, Benabou, Kelly, Street, etc. (The person (not you, Chris) who asked the question is blinkered if they think that mathematics divides up neatly into bits of independent subject areas with no interaction.)

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While, like Tim, I am not a fan of putting up barriers, I don't think us younger folk can be blamed for working within the fences that were there before us and which we don't yet have the means to dismantle... (Which is to say: I don't find Chris's phrasing so objectionable.) –  Yemon Choi Dec 8 '11 at 19:22
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@Yemon I do not object to Chris's phrasing at all since he is merely reflecting other peoples' blindness to some overall unity of mathematics. My department was shut in Bangor however because some people play these silly games of divide and rule and of putting things into boxes. It is necessary to write within 'interest groups' when writing papers of course, but don't always stick to the demarcation set down by others. The great thing about category theory journals, for instance, is that there is a very wide range of interests represented in the papers that appear in them. –  Tim Porter Dec 8 '11 at 20:44
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@Yemon mathematics is too big and important to be cut up and divided into little (vulnerable) pieces! –  Tim Porter Dec 8 '11 at 20:44
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I forgot another point worth making. Economists claim parts of mathematics as being economics as it is useful to them, and everyone accepts that mathematics cannot be useful. I fear that there may be some mathematicians who declaim that 'X is not useful' as it is not yet useful to them who are working in subject Y, but if X becomes applicable to Y, X will become subsummed into Y and can then be accepted as being useful. I therefore do not like 'useful' as it is not a well defined term. Chris's question is, however, one that does get asked frequently. (Sorry to go on about this a bit!) –  Tim Porter Dec 8 '11 at 20:51
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B-Fields in string theory form a 2-category.

This has physical relevance, for example when string theories are glued together along defect lines on the worldsheet. Along a defect line, the two B-fields are related by a 1-morphism. On a junction between defect lines, there is a 2-isomorphism relating the 1-morphisms of the various defect lines ending at that junction.

The paper Affine su(2) fusion rules from gerbe 2-isomorphisms by the physicists Ingo Runkel and Rafal Suszek explains very nicely the phyical relevance of these 2-isomorphisms, certainly far outside the realm of higher category theory.

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All examples so far involve only invertible k-morphisms for k≥3, i.e., they can be described as (∞-)bicategories. I would like to give an example of an interesting tricategory (with non-invertible 3-morphisms): The tricategory of conformal nets, as constructed by Bartels, Douglas, and Henriques.

This tricategory is used by Douglas and Henriques to give an algebraic description of string structures in the same vein as the existing algebraic descriptions of spin structures and orientations.

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