Question

What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:

• Joyals Combinatorial Species
• Grothendieck's Galois Theory
• Programming (unification as computing a coequalizer, Tatsuya Haginos categorical construction of functional programming)

I am sure that these only touch on the surface so I would be most grateful to hear of more examples, thank you!

edit: To try and be more precise. "Application" in the context of this question means that it makes use of slightly deeper results form category theory in a natural way. So we are not just trying to make a list of 'maths that uses category theory' but some of the results which exemplify it best, and might not have been possible without it.

Answer 1

One can find a couple of "concrete" striking applications of category theory in algebraic geometry. For example:

• Faithfully flat descent follows from Beck's monadicity theorem.

• Grothendieck duality, follows from Brown representability, by work of Neeman.

Answer 2

The finite vector space analog to Ramsey's theorem was proved using categories the paper is available here

Answer 3

For a while, my answer to this question was algebraic K-theory; what little I know of it, I learned from Quillen's paper, and it was a relief to finally see an example of category theory being used in an essential way to do something that was not just linguistic. Quillen defines the higher K-groups of an exact category by forming a quite different category in some combinatorial manner that seems to strip away any vestige of a connection to something non-categorical, and then taking its geometric realization and homotopy groups. The whole process: ring to module category to Q-construction to geometric realization, was the first argument I'd seen that category theory could do more than just rephrase perfectly good theorems confusingly.

(Now my answer would be "perverse sheaves", though.)

Answer 4

Noncommutative algebraic geometry(in the sense of Gabriel-Rosenberg, Artin-Zhang,Van den Berg) are based on category(abelian or Grothendieck category). They consider category as category of quasi coherent sheaves on some noncommutative space. This idea was proposed by Grothendieck and then re-quoted by Manin. Without category theory, this subject can not be built. More information is in Noncommutative Algebraic Geometry and Theories of Noncommutative geometry

Another kind of Noncommutative algebraic geometry is based on Functorial POV. It was proposed by Gabriel in the theory of algebraic group and then developed by Grothendieck in commutative algebraic geometry and then Kontsevich-Rosenberg developed noncommutative stack theory via this POV.

Noncommutative derived algebraic geometry is also based on category(triangulated category)theory.

The relevant names(maybe I will miss some of them)are Manin-Beilinson-Drinfeld, Kapranov, Deligne, Bernstein, Bondal-Orlov-Lunts,Kontsevich-Soibelman,Toen,Van den berg, Lurie, Keller,Neeman and others

Answer 5

First, a comment on `studying category theory for its own sake': this slur was very much setting up a straw man. Those accessing the category theory discussion list will know that the discussion there ranges very widely, and actually discusses issues in mathematics, in contrast to other email discussion lists I access.

Second, I have found some elementary facts from category theory very useful; examples are `left adjoints preserve colmits, right adjoints preserve limits'. Many years ago, listening to Albrecht Dold on half exact functors made me realise how I could cut down considerably a proof from my thesis by using the basic idea of representable functor: this automatically led to the existence of a homotopy equivalence making a diagram commutative. Again, the theory of ends and coends does make life simpler in discussing geometric realisations.

Third, I have fairly recently realised that the general framework of fibred and cofibred categories is specially useful for discussing pullbacks and pushouts for certain hierarchical structures with which I have dealt. A basic example here is the bifibration (Groupoids) $\to$ (Sets) given by the object functor.

I wish I had a good application in my work of some of the deeper theorems!

Answer 6

The recent developments in homotopical algebra (after 1990) would not be possible without the use of category theory, and more precisely the theory of locally presentable and accessible categories. I am talking about the theory of combinatorial model categories (model categories such that the underlying category is locally presentable).

Answer 7

This question is too vague. Off the top of my head: algebraic topology, homological algebra, etale cohomology (Weil conjectures), homotopical algebra, topological field theory, Mackey functors, Kazhdan-Lusztig theory, ...

Answer 8

The most recent book of Nick Katz [see https://web.math.princeton.edu/~nmk/mellin398.pdf ] proves extremely concrete equidistribution theorems for certain families of exponential sums. Categories enter in three essential ways (at least): (1) all the work going to Deligne's Weil II version of the Riemann Hypothesis over finite fields; (2) the theory of perverse sheaves; (3) the Tannakian formalism to recover a group from a category. In this, the new contribution of Katz in this book is (3): essentially, the equidistribution is proved using the Weyl equidistribution criterion, and all analytic estimates follow from (1). But if one doesn't know that there is a group underlying the families of sums (or rather the unitarized Frobenius automorphisms which give rise to these sums), one doesn't know what these estimates are really proving.

For more traditional families of sums, one uses instead Deligne's Equidistribution Theorem, where the group is given concretely as monodromy group of a lisse sheaf, but Katz's family are not parameterized by an algebraic variety, and the Tannakian category arises by looking at a category of perverse sheaves equiped with a suitable form of multiplicative convolution.

This is, I think, completely amazing...

Answer 9

Schlessinger's criteria and deformation of Galois representations, see e.g. Mazur's article in Cornell-Silverman-Stevens.

Answer 10

Toposes and categorical logic. ZOMG.