27 thoughts on “Opinions of Category Theory

  1. Is the fact that this entry is filed under Uncategorized a not-so-subtle hint of the opinion of the poster?

    I am a big fan of Category Theory, and have used it in some published papers. But I do not believe that it is saviour (sorry, that’s how I spell that word) or destroyer, it is just another tool. It is particularly useful when one wants to expose structure; it can be just as bad a tool, say when one is doing pure computations. I use category theory, I use set theory, I use various logics, I use type theory, whatever the best tool for the job at hand.

    Category theory does not help me at all when I am trying to figure out the long term behaviour of some function given by an ODE – so I don’t use it when I am doing that. It does help me a lot when I am trying to modularize various pieces of code, or when I am trying to give a proper semantics to a pattern-matching language, and so on.

  2. In addition to its role in exposing mathematical structure, category theory (CT) has an associated role of exposing structural similarities between different parts of mathematics. This was the reason for its development initially — to provide a framework in which different homology theories could be compared. Only CT enables us to say precisely what we mean when we say two things are the same.

    For me, however, there is another role which is just as valuable: CT provides a framework to describe the doing of mathematics itself. This is neat!

    I have always been intrigued by the tension between the fact that CT aims to abstract away from specific details of mathematical spaces and the fact that lots of the theory of CT seems to be about specific categories (the category of sets and functions between them, etc) and/or about generalizations of specific objects (eg, truth values in toposes). It’s almost as if mathematicians, even when doing the most abstract of theoretical work, can’t help talking about specific instances.

  3. I’ve been amiss in categorizing posts. I just went and fixed a bunch that were miscategorized. I’ve put up posts with jokey paragraphs and the word “Discuss” before, and people seemed to like them, so I thought I’d try it again.

    I have conflicted feelings about category theory, which prompted the comment. I thought some comments would help me clarify my thinking (which they have). I think category theory is important, and everyone in math should learn it, but it’s importance can be overrated. It shows the similarity in different parts of mathematics, but in a way that I find superficial. The “groupness” of groups is not captured by the fact that a group belongs to the category Group. You could never deduce the Sylow theorems from category theory alone: you need to think of groups as combinatorial objects, and get in there and actually count things. (I suppose this is an argument for 2-category theory, since you can make a group into a category, and treat the group acting on a set as a function into Sets, but the idea of a group as something that acts on other things is a pre-categorical intuition.)

    A couple of months ago I was reading about monads, which are defined in a categorical manner. I learned nothing. I had to rewrite it all into a non-category-theoretic language to understand it. I’m sure that for lots of people that category-theoretic language is natural, but for me it’s usually not. (In fact, when I’m not actively thinking about limits of diagrams, I now think of a generic category as being like a big semigroup.)

  4. Oh Walt, you instigator you! Why didn’t you just come out and ask if Nicolas Bourbaki was the greatest matheatician of the last century? You could also have come at it from the other angle; Poincare, hero or zero?

    I think as always the answer lies somewhere in the middle. I view it as one of the most powerful ideas in mathematics of the last 61 years. If mathematics is the science of notation of and inferencing from structure, then certainly the most powerful tool for bringing structure into high relief has to be considered one of its greatest achievements. But like any tool, it can be misused, simply because abstraction for abstraction’s sake is sterile and leads to pointless mathematics.

  5. Ok, you posted your comment as I was composing mine, so let me respond directly to your comment. Your group theory comment is a straw man. category theory is not the natural domain of discourse of the Sylow theorems (or, never underestimate the power of a theorem that counts something). Some mathematical ideas though have their NATURAL expression in the language of CT; Tensor product, Adjoints,…anything naturally defined as unique up to isomorphism. It is also a clarifying idea in terms of pointing where the mathematics should take us, as Grothendieck taught us, it is often very profitable to contruct and examine the mathematical objects that represent functors between categories.

  6. Regarding ‘counting’, category theory has a lot to say about that too! Andre Joyal invented species in 1980 for the purpose of showing how there is a natural functorial way to look at combinatorial structures, and how there is a natural functor from them to generating functions [and most combinatorics these days is done via generating functions]. I am not aware of a way to prove the Sylow theorem via category theory, but if you hop over to the categories mailing list, someone there might know.

    One of the reasons I know I personally have difficulty with some categorical arguments is my unfamiliarity with the jargon — mostly because my mathematical education used set-theoretical jargon, that is what I am most familiar with (with analysis, topology and dynamical systems being where most of my formal education lies). I am convinced that if category theory was taught earlier (say using Lawvere’s own brilliant books on the topic), a new generation of mathematicians could be a lot more comfortable with it.

    But I have also seen commutative diagrams that filled a whole page, representing a categorical proof for something that could classically (read equationally) be done in 4 lines. In that case, unless the diagram proof was showing something else important as well, the 4-line proof should be much preferred by any mathematician with a decent sense of aesthetics!

  7. Finally, a straw man in the comments! Now we can be a real blog. I’m just so proud that it was me who could be the first to introduce a straw man in order to win an argument. Actually, my intent wasn’t to win an argument, but to articulate the source of my conflicted feelings about category theory.

    While category theory is the natural language of sets with structure, it doesn’t usually capture the “thingness” of any particular class of objects, the quality that makes those objects what they are (I probably sound like Heiddeger now). Tensor product is a really good example. Tensor product can be considered as the coproduct in the category of commutative rings, as an adjoint functor of the Hom functor for module categories, as the change-of-coefficients functor from k-algebras to k’-algebras (where k and k’ are two different commutative rings). None of these capture the essence of tensor product (for me, at least), but if you understand the construction of tensor product, then most of these other properties follow quite easily. Other facts that might seem surprising (such as the fact that you can define tensor product for abelian groups, but not nonabelian groups) from a purely categorical point of view seem fairly clear when you have the concrete definition in hand.

    Jacques, what you describe is my nightmare world. Category theory has its place, but I hate the idea of force-feeding it to students earlier; to me it sounds less like education, and more like indoctrination. I know all about species, and toposes, and monads, and really, I just don’t care. It’s fine if other people care, but up to this point you don’t _need_ to care to do mathematics. If category theory had been introduced earlier in my mathematics education, I find it hard to believe that I would have gone on. I can’t be the only person who feels this way.

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  9. In response to Walt’s:

    “If category theory had been introduced earlier in my mathematics education, I find it hard to believe that I would have gone on. I can’t be the only person who feels this way.”

    For me it was the reverse. If category theory had been introduced earlier and more prominently (instead of being hidden inside a course labelled “Algebraic Topology”) I would have spent more time doing pure mathematics. Instead, I only discovered this topic at the very end of my final undergrad year, as if the faculty were afraid to reveal it to us any earlier.

    I believe the underlying issue here is how we think. Category theory matches a particular way of thinking about mathematical domains, a way based around structure. Lots of mathematicians, IME, even very good pure mathematicians, don’t think about mathematical structure, nor even think very abstractly. They think about particular problems or cases, and often in terms of problem solving.

    Look at all the 19th-century math done on limits of infinite sequences and infinite series. Most of this work was problem solving, one problem after another, each with its own unique approach and solution. Not a structure or an abstract theorem in sight. This is pure mathematics, but not as we know it today.

  10. I think that’s probably true. Whenever I read something by category afficianados, it seems clear to me that they are deriving insight from what they’re doing. I tend to not get that much out of it: to understand something I usually have to translate it out of category language into something else.

  11. While category theory is the natural language of sets with structure, it doesn’t usually capture the “thingness” of any particular class of objects, the quality that makes those objects what they are

    I think you may be looking for that quality in the wrong place: the whole idea is to abstract away from the details of the “things” and see how the systems behaves abstractly.

    This can lead to new insights and new correspondences, but it doesn’t replace e.g. group theory.

    You still need to understand the specifics of a case to see how the abstraction applies. I think most CT people take that for granted.

    Only people who don’t like/feel threatened by CT seem to think it is meant to eliminate all other approaches to math. ;-)

  12. Nobody seems to be opting for the ends of the range – saviour or destroyer – and that’s very reasonable. I wonder though if the question had been put “As you have come to understand category theory has it surprised and impressed you with its capacity to organise existing mathematics and suggest new constructions, or the opposite?”.

    There are some very interesting reprints here, including 1 and 8 by Lawvere. Even as powerful a backer of category theory as Lawvere says in 1 in 1973:

    “Of course the really deep results in a subject depend very much on the particularity of that subject and the results we offer here in the field of metric spaces, taken individually, will justly appear shallow to those with any experience. Indeed for me the surprising aspect was that methods originally devised to deal with quite different fields of algebra and geometry could yield any significant known theorems at all… But there are many particularities, for example the special role of quadratic metrics, which I do not see how could be a result of >.” p. 143

    But in the Author Commentary 29 years later he adds:

    “Thus, contrary to the apology in the introduction of the 1973 paper, it appears that the unique role of the Pythagorean tensor does indeed have expression strictly in terms of the enriched category structure.”

    I think this is a common experience that if you persevere with category theory it delivers more than you expected.

    Back in the days when sci.physics.research was thriving, they had a thread on ‘How does category theory help?’ which begins here.

  13. I hope most mathematicians continue to fear and despise category theory, so I can continue to maintain a certain advantage over them.

  14. My position about category theory is mixed, mostly because my own research is mixed. Part of my research concentrates on getting computer algebra systems to do analysis “properly” [I worked on Maple for 10 years before the lure of academia got me]. While one can use category theory to phrase the problems it has, as yet, not yielded any benefits. What seems to matter most here is the actual structure of the explicit formula you are currently manipulating. Big general theorems do not seem to be useful computationally. [Think of closed-form definite integration here]. But logic is helping, intensional logic to be precise, much better expresses the problems at hand.

    The other half of my research concentrates on how to build a good CAS system [above is more “what it means” and “algorithms”]. Here, it is nitty-gritty software construction which is the topic. And category theory has huge amounts to say about this topic! A lot of algorithms in mathematics are naturally “generic”, and this translates effortlessly to functorial semantics. When mapping back down to ways to implement things, one sees that a lot of parametrizations in computer science are pushouts and other simple categorial constructions. Of course, Goguen knew this 20 years ago, and has been telling computer scientists ever since (as have many others).

    John Baez is not the only holder of the advantage… With a student, I am mining his work (and that of his student, Jeffrey Morton) and applying it to functional programming instead of physics.

  15. David, your link to the archive of old papers is much appreciated. (Your other link is broken, or more likely the software stripped it out for inscrutable reasons of its own.)

  16. I don’t hate or fear category theory (or at least, not all of the time). I just read one piece of category theory hype too many, and I broke. (Though if the discussion had been one-sidedly anti-categorical, I would have felt compelled to defend it. I’m actually surprised I’m the only one with anything negative to say. Maybe the category of people who “hate or fear” category theory is a singleton.

    I am not new to category theory. At this point I have probably read every famous book in the subject, including MacLane, Barr/Wells (TTT), Borceux’s three volume work, Moerdijk/Reyes. My enthusiasm for the subject has gone up and down over the years, and lately it has mostly run out. Some of it is that some category theorists really do intend to supplant some existing mathematics entirely (I don’t know how else to interpret Lawvere’s plan to make category theory the foundations of mathematics). Some of it is probably just an accident of history, but in my experience the more books use a categorical language, the less they offer in terms of explaining the “thingness” of something. Some of it is that I have read categorical works that have promised more insight than I felt in the end they delivered.

    I don’t think category theory is worthless, and I wouldn’t hesistate to use it tomorrow if it arose naturally. In the field of mathematics I know best, noncommutative ring theory, category theory has been wildly successful — module categories show the subject in its best light. But to me, in the end it doesn’t offer the level of advantage claimed for it. This is probably a purely subjective judgement, in that the categorical point of view is less congenial to me than it is to some other people (or, based on the breakdown of this thread, all other people), but it’s still a considered judgement on my part.

    Jacques, I think your experience makes perfect sense. Category theory is a natural language for type theory, so the connection with functional programming is clear. It’s less natural as a language for real analysis.

  17. I do seem to have trouble posting to your blog. Is there any way you could set up a preview facility for comments?

    This time the first Lawvere quotation has been chopped off. It should

    …which I do not see how could be a result of ‘generalized

    (Perhaps the software didn’t like the French-style guillemots that Lawvere used.)

    and the link from the final word ‘here’ should be:


  18. I think one key revolution in mathematics in the 20th century was the realization that to answer the question:

    “What are the properties of this collection of objects?”

    considerable value could be got by answering instead this question:

    “What transformations of the objects in the collection leave them unchanged? “

    The second question, unlike the first, is essentially a categorical one.

  19. You forgot to mention Peter, that this is actually an insanely deep realization.

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  21. On this morning’s comments by PeterMcB and michael:

    This weblog entry at
    from Dec. 3, 2002, seems relevant.

    From the Erlangen Program
    to Category Theory

    See the following, apparently all by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal:

    * Introduction:
    * Chapter 1: Klein’s Erlangen program:
    * Chapter 2: Eilenberg & Mac Lane’s methodological extension:
    * Chapter 3: Basic principles of category theory:

    See also the following by Marquis:

    * Category Theory (article in the Stanford Encyclopedia of Philosophy):
    * Categories, Sets, and the Nature of Mathematical Entities (abstract):

  22. Peter & Michael: I’m curious what in particular you have in mind. Homology/homological algebra/homotopy, or something else?

  23. (I don’t know how else to interpret Lawvere’s plan to make category theory the foundations of mathematics).

    I think a better way to phrase Lawvere’s plan, as I understand it, is to make CT a foundation for mathematics; different approaches to mathematical formulation bring out different insights more readily.

    Any intellectual field that becomes “trendy” can sometimes generate more heat than light in the stampede to get it into theses ;-), but I agree with other posters who think that one of the fundamental insights of CT, the discovery that mapping the same mathematical idea into a different formulation can bring out a better understanding of the idea, is a profound mathematical insight.

    Having multiple equivalent “foundations” for mathematics is a natural extension of this idea.

  24. The sci.physics.research thread contains an example of the kind of thing that makes me crazy. Admittedly, it’s a second-hand report, so it could be misleading:

    At some point Joachim Lambek realized that the lambda calculus was
    basically just another way of talking about cartesian closed categories.
    More precisely, if you ask in what sort of category you can do lambda
    calculus, the answer is: a cartesian closed category.

    He told me that he presented this result at a conference on lambda
    calculus, naively hoping that henceforth the subject would be recognized
    as a small branch of category theory and basically go away. As an older,
    wiser man, he now realizes that not everyone doing lambda calculus *wants* to see it subsumed by category theory!

  25. I’ve been reading this blog for some time, but this is my first comment. I’ve found this discussion interesting because it brings back memories of my exposure to category theory in graduate school.

    First, I’m a graph theorist. I’ve always been interested in graph struture theory, mainly along the lines of the theorems of Robertson and Seymour (Neil Robertson was my advisor). I’m interested in what category theory has to say about the structure of graphs, but I’ve not found anything of note that it has to say.

    I purchased Mac Lane’s “Categories for the Working Mathematician” a long time ago, but my copy is in near mint condition. Mostly, I’ve been too busy with other things to read it, but part of my failure to study the work has been an inability to find any truth to its title.

    I first encountered categories at the beginning of an algebraic topology course. My instructor briefly discussed some of the basics, stated that category theory was useful, and then dropped it. At that time, it appeared to me that category theory was way to0 general to be able to use in the course. Plus, I didn’t understand it anyway.

    Once, Mac Lane came to give a talk. During the talk, in front of a packed audience, he stated that matroid theory wasn’t good or important mathematics, pissing off several faculty who worked in matroid theory. I found this comment to be very bizarre. Here was an advocate of a vast generalization of dubious importance dismissing a generalization of vector spaces that has tremendous importance. Was Mac Lane jealous or just cutting down people he didn’t like?

    Now, I’m not putting down category theory, I just don’t see how it is useful in my work. I find set theory to be immensely useful, so maybe, as others have noted here, my mindset is too set theoretical. But category theory seems to me to say little about everything. But maybe I just lack knowledge of category theory. Perhaps, one day, someone will show me that category theory says something important about the structure of graphs. How wonderful that day will be.

    Scott Randby

  26. Walt, I cannot speak for Peter (since I don’t know the specific examples that he had in mind when he wrote that) but the concrete Ur-example that I think of is Grothendieck’s reworking (and algebraic proof) of the Riemann-Roch theorem from a statement about pairs of varieties to a statement about morphisms between varieties. The properties of maps preserving structure become the important thing, not the objects themselves. I guess my answer is then: all three.

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