Here is an updated version of my on-going Notes on Category Theory, now 130 pp. long. I have done an amount of revision/clarification of earlier chapters, and added two new chapters — inserting a new Ch. 7 on categories of categories and issues of size (which much expands and improves some briefer remarks in earlier versions), and adding at the end Ch. 15 saying something about how functors can interact with limits. There’s quite a bit more that could be said in this last chapter, and I’ll have to decide in due course whether to expand the chapter, or return to the additional topics later, or indeed to only mention some of those topics in the end (I’m trying to keep things at a modestly introductory level). But for the moment I’ll leave things like this and move on to a block of chapters on adjoints and adjunctions. So here’s where we’ve got to:
- Categories defined
- Duality, kinds of arrows (epics, monics, isomorphisms …)
- Functors
- More about functors and categories
- Natural transformations (with rather more than usual on the motivation)
- Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
- [New] Categories of categories: issues of size
- The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
- An aside on Cayley’s Theorem
- The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
- Representables (definitions, examples, universal elements, the category of elements).
- First examples of limits (terminal objects, products, equalizers and their duals)
- Limits and colimits defined (cones, limit cones: pullbacks etc.)
- The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
- [New] Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)
Don’t hold your breath for the chapters on adjoints, though. After a very busy time for various reasons, I’ve a couple of family holidays coming up!