Application of higher categories in algebra

Question

Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categories should have great power in producing new results in algebra (K-theory, algebraic topology, algebraic geometry). What are some important applications of higher categories in these areas?

Answer 1

There are tons of such applications. For example, less than two years ago, they were used to prove the Redshift Conjecture, a statement about iterated algebraic $K$-theory. I previously wrote an answer describing the work that went into that proof (of many authors, over many years), and how $\infty$-categories were used.

Another great application of $\infty$-categories was the even more recent disproof of the Telescope Conjecture in algebraic topology, by Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank (some of the same authors as the above work). This conjecture was the last of the Ravenel Conjectures from 1984, regarding the structure of chromatic homotopy theory. The telescope conjecture was proved early on for $n=1$ and now we know it's false for $n\geq 2$, which means that the homotopy groups of spheres are even more complicated than was hoped in the 1980s. Quanta magazine covered this. What's exciting to me is that the method of disproof continues the development of our computational power in algebraic $K$-theory, and suggests we will continue to learn cool things in stable homotopy theory with these new computational tools.

In both cases, I think it's safe to say that the use of $\infty$-categories served as a powerful organizational tool, and using the yoga of $\infty$-categories severely reduced the technical hurdles required in applying the main idea to the problem at hand.

If you're willing to broaden what you consider to be "applications of higher categories," let me mention that model categories have also had major applications in algebra and algebraic topology, and experts in $\infty$-categories often tell me that these advances could be re-proven with $\infty$-categories if necessary. One that springs to mind is the resolution of the Kervaire Invariant One problem by Hill, Hopkins, and Ravenel. That's certainly algebraic topology, but perhaps more geometric than you would like. Another example is Voevodsky's Fields Medal winning work proving the Milnor Conjecture. He used a model category structure on motivic spectra, but this work could almost certainly be redone using the corresponding $\infty$-category structure (though, I don't know for sure that anyone has written down such a treatment). See also this thread and this one (which includes a discussion of Vandiver's Conjecture). One last example is the resolution of the Flat Cover Conjecture in homological algebra, via cotorsion pairs and model categories (again, the model categories could be removed today, but they were important in the moment based on what people understood at the time).