As I said in my last post, I’ve been following some lectures on category theory since the beginning of term. The only way of really nailing this stuff down is to write yourself some notes, work through the proofs, etc. Which I’ve been doing. And then I’ve done some polishing to make the notes shareable with others following the course:
So here are my current notes (50 pp.) on the topics of the first quarter of the course.
Warning: the course I’m following is for the Part III Maths Tripos (i.e. a pretty unrelenting graduate level course for mathematicians with a very strong background). My notes are easier going because I proceed quite slowly and pause to fill in all the proofs where the blackboard notes might well simply read “Exercise!”. But still, this is maths which requires some background to follow (even if perhaps less than you might think).
To be sure, I want to be thinking more in due course about some of the philosophical/foundational issues that category theory suggests: but for the moment my aim is to really get my head round the basic maths more than I’ve done in the past. Hence the notes, which maybe some others might find useful. So far, I cover
- Categories defined
- Duality, kinds of arrows (epics, monics, isomorphisms …)
- Functors
- More about functors and categories (and the category of categories!)
- Natural transformations (with more than usual on the motivation)
- Equivalence of categories (again with a section on the motivation)
Enjoy! (And even better, let me know where I’ve gone wrong and what I can improve.)
Re set theory for category theory, Michael Potter had something in the set theory in his earlier set theory book that was intended to make it easy to embed category theory. It’s mentioned on page vi of Set Theory and Its Philosophy.