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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.

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This question is too board. Category theory play roles in half of mathematics –  Shizhuo Zhang Mar 25 '10 at 16:21
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Community wiki? –  François G. Dorais Mar 25 '10 at 16:35
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@Martin: My understanding is that Joyal's theory marks the first time the combinatorial interpretation of functional composition was made rigorous, which I think is a great application. Anyway, I'd have to agree that this question is too broad. –  Qiaochu Yuan Mar 25 '10 at 16:43
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I'd like to see the question made a lot more focused. Vast swathes of mathematics, from about the Grothendieck era onwards, have used category theory. If you're not careful, this will turn into an unproductive debate about whether you call various things "applications" of category theory, or say that they use categorical tools, or "merely" use categorical language, or display categorical thinking, or whether something that was done with category theory could also be done without, etc. So for the moment I'm going to vote to close -- but if you can sharpen it, I'll cancel that vote. –  Tom Leinster Mar 25 '10 at 18:42
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Based on the title, perhaps the interpretation should be "results which essentially depend on a theorem in category theory". –  Tom Church Mar 26 '10 at 2:11

11 Answers 11

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.)

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

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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

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

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The link in this answer is broken. –  David Roberts Mar 28 '11 at 3:25
    
I have fixed the link. –  Kristal Cantwell Mar 28 '11 at 3:55
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How or why is this any more striking than Ramsey's theorem? –  Kevin H. Lin Mar 28 '11 at 5:12
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Additionally, I'm not really seeing how category theory is used. I see the word "category"... –  Todd Trimble Aug 19 '12 at 13:45

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).

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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, ...

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Sorry I have not phrased the questions quite correctly, this is not the sort of answer I was after. –  muad Mar 25 '10 at 19:26
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I agree. I think you want to avoid this type of discussion. However I am not clear on what the question should ask. –  Bruce Westbury Mar 25 '10 at 19:44
    
Bruce, your post is more or less what I would have said. –  Charlie Frohman Mar 25 '10 at 22:32
    
Muad, Bruce probably does not say only that there are applications of category theory in alg. topology, in homological algebra and so on, but that these whole branches, properly and effectively formulated and practicised are themselves applications of category theory; and deeper you go there, usually the category approach is more indispensable. For example, algebraic topology is study of functors from the category of topological spaces to some algebraic category. –  Zoran Skoda Mar 28 '11 at 13:40
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To understand muad's intended question better, consider the question, what applications does set theory have to the rest of mathematics? The knee-jerk answer is that set theory is ubiquitous so the question is too vague. On the other hand, many mathematicians (rightly or wrongly) regard the work of professional set theorists as being totally irrelevant. So I'd interpret muad's question as: If you study category theory for its own sake, will you ever prove a deep theorem with unexpected applications? Or is it, as Miles Reid claims, one of the most sterile of intellectual pursuits? –  Timothy Chow Mar 28 '11 at 15:39

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...

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Erik Meijer, at the time working for Microsoft, and his group created the dual of an IEnumerable, an IObservable, which led to the Rx Framework. According to Erik, this was an explicit use of category theory -- Erik's new venture is called Applied Duality

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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!

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

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Toposes and categorical logic.

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This would be an excellent example if only it were followed up with some detail and references. –  Mitch Harris Jul 30 '12 at 15:25

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