Are higher categories useful? - MathOverflow

Are higher categories useful?

2011-12-08T17:00:16Z

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.

Answer by André Henriques for Are higher categories useful?

2011-12-08T17:20:46Z

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.

Answer by Kevin Walker for Are higher categories useful?

2011-12-08T18:17:08Z

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.

Answer by Tim Porter for Are higher categories useful?

2011-12-08T19:04:35Z

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.)

Answer by Konrad Waldorf for Are higher categories useful?

2011-12-08T19:37:28Z

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.

Answer by Konrad Waldorf for Are higher categories useful?

2011-12-08T19:46:13Z

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.

Answer by Mike Shulman for Are higher categories useful?

2011-12-08T22:05:15Z

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.

Answer by Dmitri Pavlov for Are higher categories useful?

2011-12-11T15:42:36Z

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.