## Future directions

I’m wrapping up my coverage of ring theory (for now). There’s a lot I’ve left unsaid about rings, and also about groups. I’m hoping, though, that I’ve given a certain amount of a feel for how algebraic structures work in preparation for the next topic: categories.

There are a number of readers, I know, who have been waiting for this point almost as much as I have been. There are also some who are dreading it. Everything up until this point has been stuff that everyone has to know, but categories are still a bit controversial in some circles. Many people find them even more abstract, or technical, or even content-free than other parts of algebra.

Category theory is at turns praised and derided with the same phrase, “abstract nonsense”. Indeed the earliest uses were to make general statements about algebra, just like ring theory makes general statements about polynomials, and polynomials make general statements about numbers. For some reason there are still mathematicians who draw a line in the sand and say, “Here! No further!”, just as others saw it as the next natural step.

Personally, I have been drawn to categories since I knew they existed. I still remember being shown the natural transformation from the identity functor on the category of vector spaces over a given field to the double-dual functor, and going back to Jeff Adams’ office (yes, the same Jeff Adams) again and again for more back in the spring of 1999. I hope now to say what it is that I saw then (and still see) in category theory, and to make the case for them. I really, honestly believe that within the next quarter-century nobody will be able to get a bachelor’s degree in mathematics without a passing familiarity with categories any more than one could avoid groups now, and it’s not just due to politicking on the part of its proponents as I’ve heard asserted.

First of all, categories *are* tremendously useful as a metamathematical language. I’ll show in the future how it unifies the First Isomorphism theorems, for example. I’ll also show how, in the language of categories, direct products of groups are like greatest lower bounds.

“So what,” the naysayer cries, “if this language says that those two concepts are related?” So, mathematics is about analogies. I can begin to understand *this* because I definitely understand *that* and this and that are similar in a certain way. Maybe knowing something about greatest lower bounds will tell me something new to look for in direct products of groups. Even if not, the relationships can help illuminate to newcomers — be they students or just lay readers — the essential points of the structures we consider, and more importantly *why* we consider them.

But there’s also another side of categories that the opposition completely ignores: a category can be just as useful a concrete mathematical structure as a group can, and the framework of categories can harmoniously sew together other objects into a coherent whole. The various rings and modules of matrices over a given field meld into the category of all matrices over that field. The braid groups weave together into the category of tangles.

And what do we gain from this categorical viewpoint? If unifying language isn’t enough for you, try this: category theory is, at its core, *the* language of the analytic/synthetic approach to mathematics in particular and all sciences in general. The scientific epistemology is to break complicated systems down into simpler parts, to understand those simple parts, and to understand how to reassemble them into the whole. This is exactly what category theory brings to the table: a systematic study of the nature of composition and how compositions transform when moving from one domain of discourse to another.

Category theory is the language of analogies, and analogies are the lifeblood of mathematics. Algebra gives us analogies between equations. Categories give us analogies between theories. Our future is concerned with analogies between analogies.

Well I am looking forward to the discussion of categories anyway. I was hoping you would get round to them since the discussion about the First Isomorphism Theorems on here. As far as I am aware; I don’t think that we cover category theory at all on my undergraduate degree. When I first came accross the concept; I was quite interested in it; it seems that in principle, it could be a great pedagogical tool for teaching algebra because instead of looking at high level detail of how say, the First Isomorphism Theorem works in groups, then Rings, etc., we are interested in the very general overall principles of what it is saying about structure in algebra more generally.

Comment by Jake | May 20, 2007 |

I can’t wait for the holidays to start so I can start reading all the posts you have made on group theory, ring theory etc properly! So I’ll take your word about category theory!

Comment by beans | May 20, 2007 |

I don’t want to rain too hard on your parade, but I think it is extremely unlikely that a knowledge of category theory will be a ubiquitous requirement in undergraduate mathematics at any time in the forseeable future.

In fact, although I am not proud of it, one can get a mathematics degree at my institution, the University of

Georgia, without learning about groups! (The current

requirement is one semester in abstract algebra, which covers rings and fields and emphasizes symmetry and

geometric application.) The spirit behind this is

well-taken: most math majors do not intend to continue

on to graduate study, and many of them are intending

to use what they learn in a more practical capacity. As a young faculty member, I am tempted to push for more

“fundamentals” — analysis, algebra and topology — to be

required of all of our majors. I would not fight for category theory.

To be clear, I am not against it. You might be interested to know that category theory was embedded into the algebra

sequence I took as an undergraduate, so that I learned about forgetful functors and free objects alongside groups, rings and fields (and monoids, as I recall). I feel the same way about category theory as I do about set theory: a little bit goes a long way. Certainly a serious student of pure mathematics should learn about the categorical — and especially, the functorial — perspective at some point. The idea that one should not just consider mathematical objects individually but also systematically consider all maps between them is indeed a very powerful one, but this perspective seems more useful as a philosophy than a theory.

You write that many mathematicians find categories to be more abstract, technical and content-free than other parts of algebra. I’m not sure why you restrict to “algebra” rather than “mathematics” — surely category theory is not a branch of algebra (if anything it is a collection — but maybe not a set! — of corridors joining the various

mathematical disciplines), but let me weigh in: Do I think that category is more abstract than most other parts of mathematics? Yes, intentionally so. (Abstraction is not necessary a bad thing.) More technical? No, not at all. More content-free? Yes, I am afraid that I feel that it is. Of course some of its proponents have cited this as a virtue, e.g. that category theory makes the trivial “trivially trivial.” But it seems less worthy of study for its own sake than most other branches of mathematics.

I would be delighted if you would post something to change my mind, i.e., tell me something about category theory that as is deep and interesting as Brauer groups, Ricci flow, the Kakeya problem…

But e.g. the fact that the standard embedding of a vector space into its second dual is _natural_ is not very deep and surprising (although it is nice to know and certainly

useful): the verification is trivial. I would say the same about the Yoneda Lemma.

Comment by Peter | June 9, 2007 |

[...] I think undergraduates will need category theory In the comments to my first category theory post, Peter had something interesting to say. So interesting, in fact, that I decided to promote my [...]

Pingback by Why I think undergraduates will need category theory « The Unapologetic Mathematician | June 11, 2007 |

[...] I am above everything else a writer in the margins (and a marginal writer), I’d like to quote The Unapologetic Mathematician: If unifying language isn’t enough for you, try this: category theory is, at its core, the [...]

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