Cambridge Year

Thursday, May 31, 2007

First Exam

First exam: category theory. It was ok. Afterwards I felt like jelly (I haven't had much practice being that tense for three hours lately =) ) but it passed pretty quickly. I think I did ok: there were a couple of places where I fudged things that seemed like they were obvious but that I didn't see immediately (but which I didn't really have time to think about) but mostly I think I got it. I finished just on time: I placed the last period just as the proctor (who is called an "invigilator" here... which is pretty scary) called time.

We had three hours and needed to answer three questions out of six. I didn't know how to answer one of them (it was about monads and I'm still a little fuzzy on them) and one about abelian categories just seemed long and ugly, so I skipped those. The ones I answered are:
1. State and prove the General Adjoint Functor Theorem and use it to prove that, for any functor F: C -> D between small categories the functor F*: [D,Set] -> [C,Set] induced by composition with F has a left adjoint. --- The first part was easy, but the second part required some thinking about the Yoneda lemma and I'm still a bit fuzzy on that (mostly cause I know what the lemma means, but going through exactly what it implies involves a few too many layers of indirection for me). The bits I weren't too sure about were here... but I got all of the reasons down, I just probably needed to spend a bit more time thinking about it. The alternative to this problem was: "'Category theory is the one area of mathematics where definitions matter more than theorems.' Write an essay arguing the case either for or against this statement, illustrating your argument with definitions and/or theorems drawn from the course." It's an interesting question and one worth discussing, but on a time limit I felt that I wouldn't really have much time to write anything intelligent or interesting so I just stuck to the math.

3. (long and irritating question involving factoring functors as a final functor followed by a discrete fibration) --- I was really annoyed about this one because it was on an exam from a few years ago and I was debating doing it for practice last night and then decided that I'd rather sleep... and then I didn't get much sleep and I also didn't do the problem. But, of course, I couldn't have known.

6. Define the notion of monoidal category, and state and prove the coherence theorem for monoidal categories. --- This problem was ANNOYING. It's not hard, it just requires a very long case analysis. Three pages of writing and an hour's worth of work for about 5 minutes of thinking. I mostly did this problem because it seemed faster than figuring out the abelian categories question.

Anyway, now I feel tired, and I have another exam tomorrow. So I'm going to read some blogs and watch some cartoons and then study.