11

To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?

To put it long: I know almost nothing about category theory - but I know that I do not like solving problems and providing arguments by diagram chasing and very "high-level" arguments.
I'm currently getting near the end of my time as a student and want to start a PhD. And while I'm not lacking a skill in algebra (which has been taught to us in an entirely category-theory-free manner), and therefor assume I could manage with some amount of category theory, I'm just not a fan of "high-level" arguments. So it's more a thing of personal taste. Therefore I would prefer to specialize in an area which does not exhibit too much category-theory (ideally: none). Now the thing is, when reading around in this forum it seems to me that also in areas, in which I wouldn't expect category theory to come up, like analysis, it actually does come up ("Lipschitz categories").
Thus I would like to know, which sub(sub)fields of big fields like PDE, or number theory can be dealt with only a very small amount of category theory. Especially: Are there any subfields of geometry that are category free ? I'm currently taking a course in differential geometry and to me it seems that category theory is (in disguise) almost everywhere, since our professor constantly explains how some diagram, that proves some assertion about manifolds is actually some category-theoretic notion.

When browsing for example through the work of Terence Tao suprisingly few diagrams and mentionings of "categories" come up, so it seems to me that certain areas of PDEs and number theory may fit the bill. But this perception may be due to that fact, that currently I understand almost nothing about what I read and therefore may have missed some categorical arguments.

Side question: Are there conversely any areas of applied mathematics that heavily use category theory ? Do there exist, for example, some applications of category theory on numerical mathematics ?

user43263
  • 667
  • 25
    Why do you want to drive in a car that lacks one of its wheels??? – UwF May 07 '14 at 12:00
  • 17
    Lots of mathematics doesn't use category theory, just category theorists/those who are use categories in passing seem to be more represented online (e.g. algebraic geometers, quantum algebraists, higher geometers) – David Roberts May 07 '14 at 12:01
  • 3
    And it's not like category theory itself doesn't use "low-level" arguments. One sometimes has to knuckle down and count the number of elements in some set, or show that some poset has a small cofinal subset, or indeed, show that two expressions are equal... – Zhen Lin May 07 '14 at 12:04
  • 11
    That said, using a touch of category theory in passing in other subfields is like using algebra when doing analysis: you don't need the hardcore stuff, just some general ideas. – David Roberts May 07 '14 at 12:04
  • 1
    Most areas of dynamical systems, Ergodic theory, spectral theory, nonlinear PDEs, and probability are pretty much category theory free from what I can tell. Low dimensional topology/geometry too in most cases is more about concrete constructions (e.g. Perelman's paper) – Piyush Grover May 07 '14 at 12:28
  • 50
    The point of category theory is to clear out the fluff, so that you can see what the real nitty gritty low level' stuff is. In any case, it's a bad idea to look for a field based on points of repulsion rather than points of attraction: That is, ask yourself 'What do I most want to work on?', not 'How can I avoid category theory?'. – Keerthi Madapusi May 07 '14 at 12:31
  • 8
    @KeerthiMadapusiPera That's certainly one of the major points of category theory. Peter Freyd once quipped, "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial", which is how I interpret what you said. Anyway, I entirely agree with your sentiment. Further advice is: once you (OP) have found an area that is based on attraction, just learn what you need on a pragmatic, need-to-know basis. If you find that low doses of category theory is part of that, then you will learn a little of that too, but otherwise don't worry about it. – Todd Trimble May 07 '14 at 13:21
  • @ToddTrimble You're right of course! I should not have made it out to be the only point. – Keerthi Madapusi May 07 '14 at 13:26
  • 2
    I agree that the question 'How can I avoid category theory?' does not make much sense. However it's more or less clear what is meant by not being "a fan of high-level arguments". Lots of things in mathematics can be done that way. Also, not all of what's formally category theory is "high level", e.g. calculus of fractions is quite "low level". – Dimitri Chikhladze May 07 '14 at 14:37
  • 5
    My five cents: I think that, for most quite pure mathematicians, category theory is nothing but a convenient language; they don't really consider it as a theory, and the common name "general nonsense" is for a reason (no offense meant :) E.g., saying that the category of affine schemes is the opposite of the category of rings does not really solve any meaningful geometric problem, and saying that a cohomology theory is a functor from the homotopy category to the category of abelian groups is merely a way to save a few lines in the definition. Alas, some authors do overuse this language... – Alex Degtyarev May 07 '14 at 16:22
  • 1
    It's more appropriate to say that they don't consider as a theory whatever category theory they use, rather than category theory itself (which many pure mathematicians may not know or do not want to use, which is also fair enough). Simply saying that something is a functor won't be considered as a theory by category theorists either. – Dimitri Chikhladze May 07 '14 at 19:36
  • I don't like how the question is phrased, but anyway. These may be of interest: 1, 2, 3. – Andrés E. Caicedo May 07 '14 at 21:31
  • I used to think set theory was free of category theory, but then I saw this: http://boolesrings.org/perlmutter/files/2014/04/ElementaryEpimorphisms.pdf. – David Roberts May 08 '14 at 05:58
  • @DavidRoberts (Besides the general theory of forcing,) comparison arguments in inner model theory are one of the best places to look for connections, I would think. – Andrés E. Caicedo May 08 '14 at 06:11
  • @AndresCaicedo - I know there are implicit connections (you know how I've studied forcing), but to see a set theorist use explicitly category theoretic language and diagrams was a pleasant surprise. – David Roberts May 08 '14 at 07:24
  • (@DavidRoberts See for example this recent preprint by Viale.) – Andrés E. Caicedo May 08 '14 at 14:33
  • @AlexDegtyarev Functoriality is not a convenient language. Think for example of Brouwer's fix point theorem, the proof using the fundamental group. It is because of the functoriality of the fundamental group that supposing the nonexistence of a fix point leads to a one-to-one map from $\mathbb{Z}$ to ${0}$. – Philippe Gaucher May 09 '14 at 09:14
  • 2
    @PhilippeGaucher Just as I said: instead of explaining what is going on, it is convenient to refer to the word "functor". BTW, it is not just the functoriality; it's the fact that the identity cannot factor through the trivial group, which is way beyond general nonsense. (There obviously are categories where identities can factor through initial objects.) – Alex Degtyarev May 09 '14 at 16:13
  • @DavidRoberts That seems to me to be a really interesting "sociological" phenomenon in mathematics community. A wild thought: Maybe "algebraic thinker" are typically younger than the analysts and therefore more at ease communicating online ? – user43263 May 10 '14 at 18:16

4 Answers4

26

As a (slowly) recovering category-phobe, allow me to suggest that you change the way you think of category theory. Specifically, don't think of category theory as a "theory". A theory in mathematics generally consists of three components: a collection of related definitions, a collection of nontrivial theorems about the objects defined, and a collection of interesting examples to which the theorems apply. To learn a theory is to understand the proofs of the main theorems and how to apply them to the examples.

Category theory is different: there is an incredibly rich supply of definitions of examples, but very few theorems compared to other established "theories" like group theory or algebraic topology. Moreover, the proofs of the theorems are almost trivial (the Yoneda lemma is one of the most important theorems in category theory and it is not even called a "theorem"). A consequence of this is that you don't have to sit around reading a category theory book before you make contact with the language of categories: the very act of understanding how people express results from "ordinary" mathematics in the language of categories and functors is learning category theory.

So now I'll try to answer your question. It is possible to work in nearly any area of pure mathematics without much category theory, and most areas have people everywhere on the category theory spectrum (with the possible exception of algebraic geometry, wherein the language of derived functors is basically built into the foundations). Analysis in particular seems to be somewhat resistant (but not immune) to categorification, and if you are really committed to avoiding categories then you might consider exploring the more analytic aspects of what interests you (e.g. geometric PDE's or analytic number theory).

But before you make that commitment, try to find some examples of categorical language in action in what you already understand. The sheer ingenuity of it all might change your mind.

Paul Siegel
  • 28,772
  • 12
    I like the spirit of this response. With regard to the theorems being almost trivial, I partly agree, but at times it can be like what Deligne said about a typical Grothendieck proof: that it consists of a long series of trivial steps where “nothing seems to happen, and yet at the end a highly non-trivial theorem is there.” – Todd Trimble May 07 '14 at 22:06
  • 1
    Not thinking about category theory as a "theory" really made it click for me (even though that indeed wasn't what I was asking :). When you say category theory comes up in analysis you mean that in comes up only in the "abstract" branches of analysis, like topolgy or functional analysis or also in the more "concrete" ones like real analysis ? – user43263 May 10 '14 at 18:06
  • [cont] And is there a "deeper reason" - whatever that would be - that analytic aspects can't be described using category theory ? Since this could be used as a foundation I probably should say "can't immediately or intuitively be described using category theory" since as a foundation if we can describe a set categorically we could theoretically describe everything. – user43263 May 10 '14 at 18:11
  • Roughly echoing Todd: What a top-level musician or athlete does looks "trivial". I am not sure what the opposite of "trivial" in this context should be -- "contrived", maybe? Why do proofs by mathematicians who have not been influenced by category theory look so contrived? The answer is in the question. When you really understand the structure of a proof, it can be broken down into "trivial" lemmas, but the mastery lies in the choice of lemmas, not in their proofs. – Paul Taylor May 11 '14 at 08:35
  • 2
    @PaulTaylor: To further illustrate my point, compare the Yoneda lemma to, say, the h-cobordism theorem in topology. A bright student with little background can learn the proof of the Yoneda lemma in an afternoon, but s/he will continue to be surprised by its consequences years later. But even a bright student with a few topology courses might need a semester to learn the proof of the h-cobordism theorem, even though the statement is elementary. As far as I know there is no theorem in pure category theory that takes a semester to learn. (Though there are definitions with this property!) – Paul Siegel May 11 '14 at 12:37
  • It took me maybe ten years to begin to appreciate the importance of the Yoneda Lemma. Category theory is not a bunch of definitions and theorems but a way of thinking about mathematics. It deepens throughout your career. My analogy with musicians and athletes is appropriate. Actually, there are very few definitions and fewer theorems, they are just very powerful. It's the mathematical equivalent of living in a forest equipped only with a sharp knife. – Paul Taylor May 11 '14 at 12:46
  • 1
    @user43263: There are ways to express even basic things in analysis, such as the spectral theorem or the Lebesgue integral, using the language of categories. But many of the hard theorems in analysis boil down to subtle estimates which (so far!) have not been simplified by clever categorical arguments. – Paul Siegel May 11 '14 at 12:51
  • @PaulTaylor: My original purpose in distinguishing category theory from other branches of mathematics was to suggest that the OP would better appreciate the subject by learning from examples without feeling obligated to spend a semester reading a category theory textbook - perhaps you would agree? Pursuing your analogy, to learn music you should start by playing, not by reading music theory books. – Paul Siegel May 11 '14 at 13:04
  • 1
    @PaulSiegel: we are expressing different, not conflicting, points of view. Actually, as a post-categorist, I tend to roll these things together with constructivism: new versus traditional. Of course there are hard estimates in analysis and similar things in other subjects, but the purpose of well-organised mathematics (eg using category theory) is to sort out the genuinely hard things from the overall management. A long complicated argument crammed into the proof of a single Theorem, undivided into Lemmas, is not a sign that the material is deep, just that the author has no bigger picture. – Paul Taylor May 11 '14 at 13:57
  • 1
    @PaulSiegel Do you know a reference where the Lebesgue integral is expressed in the language of categories ? This sound incredibly interesting (and fearsome!) to me. – user43263 May 12 '14 at 16:06
  • @user43263: for a categorical description of the Lebesgue integral, see here: http://www.maths.ed.ac.uk/~tl/glasgowpssl/ – Tom Leinster May 20 '14 at 12:50
  • 1
    I like the notion of category being different than usual theories due to not having much theorems, and them having easy proofs like the Yoneda lemma. Still, I don't agree fully, because the proofs of theorems being easy is different from the theorems being easy to prove. As been said, there quite some hard definitions, and then there are many theorems along the lines of "if ... has an adjoint ... then ... (co-)product is preserved" and those are hard to prove because you really needs to have some internal image of the things like adjoints and limits which you work with. – Nikolaj-K Dec 11 '14 at 12:22
23

I would upvote Keerthi's comment multiple times if I could. Just find an area of mathematics that makes you smile and brings you happiness. If in the course of doing research you find that you need to deal with some category theory (say to study an important paper in the area), then you'll simply deal with it as it comes up, and at least you'll have some inherent motivation to do so. (And who knows? You might wind up finding some of it clean and attractive after all, and you'll have learned something to boot.)

I can't resist adding something though, speaking as a category theorist. When I'm in the thick of "doing" some category theory, I'm almost never thinking "whoo, this is high-level stuff". Mainly, doing category theory just feels like doing a form of algebra; it's a very ordinary sort of activity. It can be very nice to realize, afterwards, how widely applicable this piece of algebra may be, but in the moment we're not thinking how mind-blowingly abstract it all is (and to me it just doesn't feel more abstract than other forms of algebra).

We all have our tastes. I'll own up that I'm not a very geometric or analytic kind of guy -- I'm not greatly experienced in those modes of thought, but I'd hate for that to turn into the type of negative attitude so familiar from cocktail parties, where you meet someone who tells you (hearing that you do mathematics) that he always hated the subject, and seems almost proud of the fact. If on a given occasion I felt that I needed to know more geometry, then I hope I would simply learn what I needed to know, and not agonize over it.

Edit: Andres Caicedo's comment under the question reminded me of something John Baez once wrote. (It's maybe a little snarky, but effective in a way Baez is renowned for.)

Todd Trimble
  • 52,336
  • 1
    You probably became a category theorist because you have an advantage there from finding category theory no more abstract than the rest of algebra. Most mathematicians who don't become category theorists (i.e. most mathematicians) probably have the opposite feeling. –  May 07 '14 at 15:32
  • 7
    Actually, I was drawn to category theory (around when I was 18) because it seemed to have answers to a lot of questions I had, like "what is the commonality between a quotient group in algebra and a quotient space in topology?" or "what is duality?", etc. And I think also in the beginning that I was enthralled by the generality and abstraction. But what I mainly want to emphasize now is that if category theory still has a fearsome reputation, then it probably shouldn't: people should think of the inner techniques as no more of a big deal than a lot of things in algebra. – Todd Trimble May 07 '14 at 16:43
  • 7
    I thought you became a category theorist so that you could be the person that defined tetracategories, http://math.ucr.edu/home/baez/trimble/tetracategories.html – Andrej Bauer May 07 '14 at 16:45
  • 5
    @AndrejBauer Yes, I'll probably never live that down. ;-) – Todd Trimble May 07 '14 at 16:51
  • Suddenly I have an urge to see someone from tex.stackexchange.com code up your definition in LaTeX...as an A0-sized poster (or two). – David Roberts May 08 '14 at 00:00
  • 3
    @DavidRoberts In the meantime, there's this: http://arxiv.org/abs/1112.0560 – Todd Trimble May 08 '14 at 01:46
  • Todd - forgot that! (The other daydream I had was that someone formalised your notes http://math.ucr.edu/home/baez/trimble/polyhedra.html and then your definition. That has certainly not been done!) – David Roberts May 08 '14 at 04:55
  • Ironically, I like category theory because it is utterly unrelated to the concept of number (not that I am bad with numbers, just that I think they are overrated.) For example, one of the first examples of a group is the integers. One of the first examples of a set is the set of natural numbers. One of the first examples of a Category is the Category of Categories. – Christopher King Jan 13 '15 at 01:24
  • @PyRulez I don't think of category as unrelated to numbers (or as unrelated to anything else in mathematics!). I guess I'll leave it at that, except to add that the phrase "Category of Categories" always reminds me of the Nicene Creed ("God of God, Light of Light, very God of very God, ...") or of Holy of Holies, etc. ;-) – Todd Trimble Jan 13 '15 at 01:33
  • @ToddTrimble I mean the category theory is no way a generalization of numbers, or numbers aren't more important than other mathematical concepts (except when they deserve too.) It seems every theory (except for set theory) is either a generalization of numbers or built with numbers (or their generalizations.) – Christopher King Jan 13 '15 at 01:37
6

Regarding your side question, more "applied" stuff that uses categories or category theory is usually in the field of computer science.

In fact a general category theory conference may have just as many people doing theoretic computer science as say people doing algebraic topology.

The more applied theoretic computer science research that I've seen are people using categorical ideas in programming, databases, and quantum computation/information. I've even seen category theory in papers in cognitive science. However, I am not aware of any work using category theory in the numerical stuff.

Dimitri Chikhladze
  • 2,302
  • Could you please give me a reference to a paper using category theory in cognitive science ? I can't image how this application of category theory could really make anything for a cognitive scientist clearer since (in my naive opinion) it would only add additional terminological baggage for someone who (I would again naively think) doesn't operate with mathematical objects so diverse that a very abstract description, as category theory provides, would help him. – user43263 May 10 '14 at 18:09
  • I don't remember the particular paper that I saw. But googling "cognitive science category theory" gives some results. I can not talk for the authors, but it seem to be related to formal languages and logic, usage of which in cognitive science might be imaginable. – Dimitri Chikhladze May 10 '14 at 19:06
5

Modern analysis is one large area where category theory has fairly minimal impact. Occasionally you see statements like "the category of locally compact Hausdorff spaces is dually equivalent to the category of abelian C*-algebras". I'm sure I could think of examples of more substantive uses of category theory, but they're pretty rare.

As a teacher, I've occasionally seen students with a background in category theory seemingly hindered by intuitions that don't work well (e.g., expecting the tensor product of Hilbert spaces to have the obvious universal property).

However, I agree with Todd's comment. If you ever find yourself in a situation where you need category theory for something you want to do, you are likely to suddenly find it much easier to pick up. At least, that's been my experience with other subjects that I thought I didn't like, until one day I needed them.

Nik Weaver
  • 42,041
  • 2
    I can't speak with authority here, but nuclear spaces might be another exception that proves the rule about categorical thinking having fairly negligible impact on analysis. (Speaking of tensor products.) – Todd Trimble May 08 '14 at 03:56
  • @Todd: how so? I don't know so much about nuclear spaces. – Nik Weaver May 08 '14 at 04:29
  • 2
    Oh, here's a serious example: Morita equivalence in C*-algebra. – Nik Weaver May 08 '14 at 04:30
  • 2
    It's not something I have detailed knowledge of, so let me further qualify it as a hunch that the area seems amenable to further categorical analysis. Chapter 5 of these notes by Taylor http://www.math.utah.edu/~taylor/LCS.pdf talks about various categorical aspects in terms of projective limits, interactions between various tensors and homs, properties of the categories of nuclear Frechet spaces and their strong duals... to a category theorist this looks rather inviting for more investigation. – Todd Trimble May 08 '14 at 05:01
  • Okay, interesting. – Nik Weaver May 08 '14 at 05:50
  • I can't help mentioning, in response to Nik Weaver's second paragraph, that I've occasionally seen people with PhDs and a background in category theory led to say irretrievably false things based on the intuition that there is only one reasonable tensor product of Banach spaces. (In fairness I regularly think completely wrong things when I read about group schemes, based on the intuition that groups have points, so each to their own) – Yemon Choi May 12 '14 at 06:27
  • @YemonChoi I know it's been some time, since this question was active, but somehow your remark that thinking groups have points is in some cases bad intuition stuck with me. Since for example geometric group theory is about viewing groups as collections of points, this intuition can't be so bad. Could you give me an example concerning group schemes, where this intuition leads you astray ? – user43263 May 19 '14 at 08:14
  • 1
    @user43263 the product in the category of schemes, with which you define a group scheme, is not as straightforward if you are thinking of a scheme as a ringed topological space (hence consisting of some underlying points with extra stuff). Thinking of schemes as sheaves on the opposite of the category of rings, it's easy, but you don't use the underlying set this way. – David Roberts May 19 '14 at 10:33