What should be the next step in studying category theory?

This is a soft/educational question and I'll flag it to be made community wiki.

A little bit of background, first. I am in my last undergraduate year, and I took a graduate course in category theory; this class really inspired me in a mathematical sense: I'm considering studying this at a graduate level and possibly doing a PhD in category theory.

The class covered elementary category theory up to elementary topoi/Grothendieck topoi and abelian categories (if you'd like to know the program more in detail I'll add it at request).

My question is essentially this: is my experience (or lack thereof) with this subject too narrow to understand the kind of decision that I'm making? I mean, I love the generality and the concepts studied in category theory because they give me a different approach to other subjects I'm studying (elementary differential geometry for example); I'm uncertain, though, as to what will be coming: I don't really know what studying category theory at a higher level involves.

Also, I'd love to know what books I should read to get a better insight of the subject. The book I got for class was Categories for the Working Mathematician and also (for the topos theory part) Sheaves in Geometry and Logic.

I really hope I made myself clear, but if I didn't just comment and I'll try to explain better.

I think there is some kind of periodicity in people asking this kind of questions :)

Nevertheless, I don't get the point of your questions, especially in

I don't really know what studying category theory at a higher level involves.

I mean, if you really studied some CT, you've already seen that its language subsumes practically every kind of Mathematics (or even some parts of Physics) you can study or imagine to study; you can turn general theorems into Differential-(Algebraic-) Geometric results, you can give the most general setting in which develop homological algebra, you can do Homotopy Theory at your favourite level of abstraction, you can use (Mac Lane & Moerdijk do so) sheaves in Logic, you can see abstract and universal Algebra from a wonderful perspective... So there's no way (for me, other people surely have a better intuition of what Category Theory "is") to give you a unique answer to this question. The good side is, you see that you can do fairly everything if you study "CT at higher level", just as you can potentially speak every known language if you "learn" basics of moving your tongue.

I see from your profile that you're an italian student from Milan; I'm not very far from there, living in Padua. It seems to me that Category Theory (as a branch of Mathematics per se), in Italy, is not really developed (to tell the truth, I seriously hope somebody will contradict me!). It is deeply intertwined with two of the topics I mentioned before, namely Logic and Algebraic Geometry; analysts don't get the point in "categorifying" their thoughts, physicists hardly know the definition of a sheaf (in my experience, even of a smooth manifold). In my opinion this is due to the fact that CT "can hardly walk alone" its path, it needs to give insights to other branches of Mathematics, as it needs to pick up examples in Mathematics seen as a whole subject.

As a (highly non-genius) student labouring from some years on the topic, the most useful advice I can give you is: He who loves Category Theory thinks Mathematics as a single, huge subject: so, if you want/can do it, learn the more Mathematics you can see, in the more wide range of topics you can reach: then Category theory will give you lots of insights. If you jump over this step, I think you'll feel like a blind man seeking the colour of the grass.

Sorry for this long stream of consciousness: in fact any answer I can give you highly depends on your answer to the following question:

Why do you like CT? Do you want to use it to answer other mathematical questions (e.g. in Geometry, Topology, ...)? Do you want to reach via its language a more precise idea of "what Mathematics is" (for example enlightening your view of Logic or foundations of Maths)?

Good luck!

• Thanks for your answer! It's great in the sense that it reaches an unsaid point in my question, which is "should I study category theory along / after other subjects in order for me to understand it (CT) better?". I also agree to the periodicity of these kind of questions, I thought I'd ask here because my professor is unavailable at the moment (plus, different opinions are quite better for judging).
– Andy
Commented Dec 27, 2011 at 14:14
• You're welcome! Just as a curiosity, are you considering to enrol in the AlGaNT Master program? Milan is a good place (if you are -like me- more Category-Theory-oriented than P-adic-analysis-oriented, surely better than Padua!) to spend your first year of master. And again, what will you study in your undergrad thesis? Commented Dec 27, 2011 at 14:50
• I'm not up to ALGaNT standards, unfortunately. Also, the undergrad thesis changed lot for my year: now the commission decides three possible subjects, will communicate them to me a week before the dissertation and I will have to choose between those. They'll also provide a list of books to study. If you want to know more contact me at my email (I think it's visible on my profile)
– Andy
Commented Dec 27, 2011 at 15:18
• @tetrapharmakon: Regarding ‘a blind man seeking the colour of grass’, perhaps a more apt metaphor here would be ‘the four blind men describing an elephant’... Commented Dec 27, 2011 at 15:23
• @Zhen Lin: yes, I had that in mind just after posting but it was not worth a correction. ;) my metaphor goes to the point too. Commented Dec 27, 2011 at 15:27

I strongly doubt that I will be able to say more then what tetrapharmakon said, nevertheless I want to say something else.

If you look a little bit you'll see that real a category theorist are also algebraists, or algebraic topologists, or algebraic geometers, or logicians,...

That's not a case because categories are first a way to think before being a mathematical object: the point adoesn't exists,ll mathematics is about studying categories, algebraists study categories of algebras (here algebra is intended in the broader sense), algebraic topologists use model categories to treat homotopy in abstract way, logicians model theories and related interpretations via categories and functors.

The point is that category and related concepts are to general (similarly to topological space) thus in order to produce interesting results one need to focus on particular kind of categories (eventually proving results in the more general settings when it's possible), this particular kind of categories were born from concrete problems in other areas of maths (mostly topology, geometry, algebra and logic, but not only).

So rephrasing what tetrapharmakon said studying category theory is studying mathematics in general, it's not a case that many of the category theorists are some sort of know-it-all of maths, as Lawvere.

I hope this answer could be of help.

• If Lawvere, of all people, is not a ‘real category theorist’, then I don't know who is. Commented Dec 27, 2011 at 15:05
• I corrected a couple of grammatical errors in your answer. Be careful when you talk English in a native commutity :) Commented Dec 27, 2011 at 15:13
• @tetrapharmakon: *speak ;)
– Andy
Commented Dec 27, 2011 at 15:20
• @ZhenLin: I absolutely didn't mean that. I apologize if it seems so, probably I haven't made myself clear. What I meant is that category theorist are usually people that know lots of thing from many different maths' branches Lawvere is an example in this sense: it dealt with algebra (for instance in his thesis), logic (see elementary topos), analysis (he defined metric spaces via enriched category theory) and we can go on. Commented Dec 27, 2011 at 15:20
• So... if categorical thinking is foundational and everyone does that, the same argument can be said on set theory. Therefore every mathematician does both category theory and set theory. Doesn't that make all those ETCS guys recursive? Commented Dec 27, 2011 at 15:24