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.
Best Answer
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:
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.