# Category theory sans (much) motivation?

So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I started to learn about the categorical way of thinking.

I think that the deal is as follows: One begins to gain all sorts of intuition about, for instance, groups when one realizes that it doesn't pay to think about elements of a group nearly as much as it does to think about morphisms to/from a group. The category-theoretic point of view is a tool that lets you gain intuition by moving up and down the hierarchy of abstraction. Maybe I'm not articulating this that clearly, but hopefully you've had similar experiences.

The problem is, I don't know any way to get a handle on the categorical way of thinking without learning category theory, and I don't know any way of learning category theory without wading through tons of abstract nonsense before you can begin to understand why it's valuable. (This is an even worse problem if, like my friend, you don't have any interest in the motivating examples from topology or geometry.)

So, what can I recommend that might help my friend start thinking categorically without drowning him in a sea of abstraction? Or is the "algebra sucks" phase a necessary stage of mathematical development?

ETA: Just to be clear, this is mostly undergrad-level stuff we're dealing with, so while I'm not opposed to easy ways to motivate category theory or get someone hooked, the fewer prerequisites the better...

• For what it's worth, I never had an "algebra sucks" phase because for me, at least in high school, the big motivation for algebra came from number theory, and I've always been interested in number theory. – Qiaochu Yuan Oct 19 '09 at 3:08
• Yes, I also have been in love with algebra since I first learned the axioms of group theory. – Harry Gindi Apr 15 '10 at 13:34
• This may sound a little contrarian, but is category really the right way to go for this individual? It seems to me that a student struggling with algebra would need less abstraction not more. There are a number of concrete problems in number theory and combinatorics (in how many ways can one label the sides of cube?) that can be used to motivate groups and rings. – Donu Arapura Apr 15 '10 at 19:12
• I've loved algebra ever since working though Herstien's TOPICS IN ALGEBRA in my honors undergraduate algebra course a few years ago. But I agree category theory at this stage is highly questionable at this point and is probably what's confusing your friend.Category theory is all about The Big Picture and that Picture is constructed from many,many examples.THAT'S what your friend should be concentrating on. – The Mathemagician Apr 15 '10 at 21:13
• @Donu: I think it would depend on the person. Sometimes, especially at the undergraduate level, more abstraction is actually easier, because it strips away the possibly confusing details and leaves the bare essentials. For instance, proving that any continuous function $f: [0,1] \to [0,1]$ must have a fixed point is a lot easier than to be given a monstrous (or even not so monstrous) function which satisfies those hypotheses and having to prove the existence of the fixed point in that particular case, because the first case is so sparse there is not much we can attempt to begin with. – Thierry Zell Nov 7 '10 at 0:23

There are two books that I would highly recommend to any one (especially a high school student or an undergrad with some mathematical maturity) wanting to learn category theory (CT) without necessarily learning all the "abstract nonsense." That is, it is possible to become thoroughly acquainted with a categorical way of thinking, if you will, without learning all the definitions beforehand. The books are "Conceptual Mathematics: A First Introduction to Categories" by Lawvere and Schanuel, the second being "Sets for Mathematics" by Lawvere and Rosebrugh.

The first book (Conceptual Mathematics) is, I believe, the best introduction to CT via sets and functions. From a pedagogical point of view, it is very well-written with lots of examples that slowly but surely develop the diligent student's category-theoretic intuitions. The notion of a functor is only mentioned in the middle of the book and that too in the context of a monoid (which is a very nice concept to learn) and that the notion of a natural transformation isn't even mentioned in the book. Indeed, universal mapping properties are introduced two-thirds of the way into the book. However, that does not prevent Lawvere (and Schanuel) from talking about a lot of stuff. One is surprised by how much CT one learns by the time one finishes reading (and solving the exercises in) the book.

The second book (Sets for Mathematics) essentially is a superb exposition on the elementary theory of the category of sets, and the book can be read either after finishing the first book (mentioned above) or in conjunction with it after one has gained some proficiency in understanding and writing (simple) proofs in CT.

Anyway, if your friend really wants to learn CT without getting "bogged down" by all the definitions, then Conceptual Mathematics is the book to read and absorb. The book looks deceptively easy and devoid of a lot of the standard category theoretic content, but nothing could be further from the truth!

Hope this helps.

• The two books you recommended look very promising. I've just started looking at them. Looks like they contain the motivational background I've been looking for. – John D. Cook Nov 3 '09 at 17:08

The Unapologetic Mathematician covers category theory before it covers group theory, so after a little more experience with abstraction you might want to expose your friend to the beginning posts. I also like Todd Trimble's series on basic category theory here, especially his version of the Yoneda lemma for posets. But all of this might still be a little too abstract.

When I first started thinking about category theory I was very interested in products and coproducts. Until you learn to differentiate Grp from Ab, the distinction between the direct product, direct sum, and free product gets confusing, but the category-theoretic perspective clarifies this a ton. But the example that made me really excited was realizing that the product and coproduct in a poset are just the supremum and infimum. I was excited that two different natural definitions in two different subjects could be thought of as special cases of the same even more natural definition.

Anyway, it might be a good idea to ask your friend why he/she hates algebra. It might just be that your friend hasn't been exposed to a lot of concrete examples, especially if your friend happens to be an MIT student taking Artin's course.

• Hahaha, nope, not the case. This is undergrad algebra, which I normally wouldn't think about talking about category theory but I'm pretty sure he can handle it. (Although I do have some algebraic-geometry-dear-god-why-are-there-no-examples questions...) – Harrison Brown Oct 19 '09 at 3:08
• A little off-topic, but if you want a lot of examples in algebraic geometry you should take a look at <a href="books.google.com/… table of contents. – Qiaochu Yuan Oct 19 '09 at 3:22

I had a friend who once told told me, "Start thinking in terms of universal properties and exact sequences. Just do it." I trusted him, and this way of thinking quickly warmed me up to the generality of category theory :)

Frankly,I think it's crazy to try and learn algebra cold turkey categorically. It's kind of like defining a Banach space on the first day of calculus. At top research schools,though-professors have a lot of trouble,especially if they're young hotshot researchers,coming down to the level of beginners. I'm sure to someone like that category theory is obvious to anyone with half a brain.But in the real world,not so much. The best book I can recommend for your friend-and I'd recommend it to anyone needing to learn category theory on thier own without much mathematical training-is Category Theory By Steve Awodey in the Oxford Logic Series. Awodey was a PHD student of Saunders MacLane who learned category theory from the man himself and his classic Categories For The Working Mathematicain,which is THE introduction for mathematics graduate students.His purpose in writing this book is to write "a book for everyone else". I think he's MORE then succeeded here. It's kind of expensive,but worth it. Another book you might want to check out is Paulo Aluffi's "Algebra:Chapter 0". This beautifully written algebra text gives algebra from a categorical perspective while keeping it fairly gentle and concrete-your friend might find it just what he needs. Lastly,tell your friend examples and motivation are the lifeblood of mathematics and he shouldn't listen when people give him "The Emperor's New Clothes" party line.

I remember the first thing I learned which sold me on category theory: the arrow-theoretic definition of injections and surjections. Thinking about how to define an injection without being allowed to say "element" is a good, and possibly fun, exercise.

Besides this, +1 on Qiaochu's comment - realizing what "coproduct" means in the category of groups vs the category of sets is another great "aha!" moment.

I received my first glimpse of category theory (well, first glimpse after a friend's excited ravings about functors) from reading the first chapter of /Arrows, Structures, and Functors: the categorical imperative/ by Arbib and Manes. As I recall, it recasts basic set theory in the language of categories, and it's quite clear and readable with minimal prereqs. The rest of the book doesn't have much in the way of prereqs either but for some reason I didn't read much more of it until I was significantly older and more mathematically mature. This may in part be because the book takes a while to get around to doing functors. (Part of it also is that I soon realized that category theory was the study of arrow diagrams. My first exposure to abstract algebra, by self-study from Mike Artin's book, had induced in me a dislike of arrow diagrams -- some time later I realized that arrow diagrams are my friends, but Artin's book didn't encourage that sort of thinking.)

More generally, you may find introductions to category theory aimed at computer scientists (the book I mentioned above is one such) to be helpful.

This might sound kind of nutty, but my first real introduction to category theory was a 2-week summer course on 2D topological quantum field theories (the notes for which, minus the actual category theory background, appear to be available here).

I think the great thing about this approach was that it did not require learning much heavy categorical machinery but was still able to demonstrate the power of categorical thinking. Having the cobordism category immediately frees you from thinking that a morphism is necessarily some sort of map. And the overall result, that we can get some very nontrivial algebraic structure (a commutative Frobenius algebra) out of some very natural topological structure, is beautiful and quite striking if you're not used to thinking about the connections between disparate areas of mathematics in such a manner.

Of course, I'm probably not a good poster child for this approach given that I spent the next few years thinking I wanted to be an analyst. (Possibly if somebody had been able to explain anything at all to me about the "Q" in "TQFT," this could have been avoided.) But the course definitely stuck with me and motivated me to explore the world of categories more deeply later on.

• If you wanted to free yourself from thinking that morphisms are functions, a much easier choice is the category of sets and relations. – Qiaochu Yuan Oct 19 '09 at 4:13

I've found a major stumbling block when trying to introduce category theory to new people is a tendency to think of objects as "things" instead of "types of things". So, after coping just fine with definitions of the category of sets, groups, etc., when asked to build their own category, one gets things that miss the point entirely.

Often these first attempts look more like finite state machines than categories, which is perhaps because the use of diagrams in this case misleads. It makes a person think there is some spacial or temporal separation of objects in a commutative diagram, rather than a logical separation, namely a distinction of type. Thinking of categories as a typing system (esp. coming from a CS background), was my first "ah-ha" moment.

Of course, whether you can say the work "type" without thinking the word "set" is another issue.

• Coming from a CS background, I had a lot of trouble with category theory until I stopped thinking of objects as types. The type/term distinction is a much harder line than the object/morphism distinction. Eg, comma categories, arrow categories, hom functors, etc. all blur the line in ways I as a type theorist found terrifyingly structureless. This is, of course, exactly the opposite complaint of typical mathematicians, who say CT is far too rigid, and goes to show you can't please everyone! – Neel Krishnaswami Apr 15 '10 at 14:00
• How does hom functors blur the type/term distinction? The hom functor takes your source and target types S and T to the type of a function with input S and output T. Or is it that you're used to not allowing functions full citizenship? – Mikael Vejdemo-Johansson Apr 15 '10 at 14:48
• No, it's the way that they use of the impredicativity of set theory. The construction of exponentials in the functor category $[\mathbb{C}, \mathrm{Set}]$ illustrates this: $(F \Rightarrow G)(X) = \mathrm{Set}(\mathbb{C}(X, -) \times F, G)$. See how you use the hom-functor of Set to define a set? That's a really daring definition to a type theorist, since you're using something "big" to define something "small". If you like, you could say categories clearly exposed my discomfort with sets as opposed to types. – Neel Krishnaswami Apr 20 '10 at 10:41

The book Frobenius Algebras and 2D Topological Quantum Field Theories by Joachim Kock sticks to the concrete and geometric point-of-view while supposing very little higher algebra (see chapter 2). At the same time, it motivates the study of higher category theory quite well. It would help your friend to keep this picture in mind when studying more technical things.

So your friend hates algebra and doesn't have any interest in the examples from geometry and topology...

Try physics: An algebraic quantum field theory is a functor from some category of manifolds to C^*-algebras, the composition perservation property has physical meaning (that things are determined locally, somehow, don't ask me...)

Or analysis: F.W. Lawvere, Metric spaces, generalized logic, and closed categories, TAC Reprints (http://www.tac.mta.ca/tac/reprints/)

There also are works by Reinhard Borger (FernUniversität Hagen) on category theory and functional analysis.

If your friend doesn't have interest in these either then leave him in peace. One can solve partial differential equations without categories, after all. It's hard without some abstract algebra though...

I always think it's best to have examples to work with. The examples I was first exposed to were in algebraic topology, but I think there's an enormous store of examples in the world of elementary functional programming.

Objects, morphisms, (endo)functors, natural transformations, products, coproducts, limits of functors, colimits of functors, monads, ends, coends, representable functors and so on all have not-quite-trivial but natural interpretations as programming constructs. These examples don't require a vast programming knowledge to understand.

The important point is to bear in mind that these are just specific examples and that the categorical definitions are, of course, more general. But often proofs written about functional programs generalise correctly to proofs in general category theory with minimal change (because various fragments of various programming languages are in fact internal languages for various types of category).

Maybe you could construct an explicit exact sequence which describes something irl about which your friend has already an intuition?

Start with a tiny category (1,2,3 elements)

which is theoretically simple, and by a sequence of algebra-respecting surjections how a complex model of the real-world thing can be broken down (at different levels) into smaller bits with an easier logic to them. (Alternatively you could branch out in different directions and show how chain 1 captures this aspect of the model, whilst chain 2 captures that aspect of the model.) I don't have a ready example for that right now but I'll edit this answer if I think of one.

You could also spend time talking about "congruence"—just like it would be idiotic—pedantic beyond the worst human pedant—to treat two different hand-writings of the letter a as having different meanings, we want to be able to use the word "the same" in mathematics as we use it in normal language—like, "OK, not the same same, but, you know, basically the same". (And this needs some precise definition in order to give us, in mathematics, the freedom-of-movement we get gratis in eg English.) I like the familiar toddlers' play blocks as a metaphor for "The same like how?".

I'm not sure if this applies to your friend, but the simplest functor I can think of is between $(o,e,\mathbb{Z}_+) \longleftrightarrow (pos,neg,\mathbb{R}_{\times})$. Anyone who remembers grade-school arithmetic can follow the technical aspects of that relationship. (And you cover "the arrows changing along with the objects"—the relationship wouldn't stay the same if I substituted positives for evens without changing + to ×—and I think this is intuitively clear to anyone whose brain has been sufficiently jogged.)

I'm not sure if that captures what took you out of algebraic doldrums, but that's how I would explain category without busting out a Rube Goldberg of definitions first.

I learned basic category theory from some good book (I forgot which one) and more advanced stuff from Gelfand-Manin. I never encountered the problem of I don't know any way of learning category theory without wading through tons of abstract nonsense before you can begin to understand why it's valuable.

So my suggestion might seem trivial, but keep looking for a good book!