# Notes on Category Theory v.4

I wasn’t able to do much work in January for family reasons, but levels of concentration and energy are returning. So, much later than I’d hoped, here at last is an updated version of the Notes on Category Theory (still very partial though now 109 pp.). There are three newly added chapters, and some minor tinkering earlier. We now cover:

1. Categories defined
2. Duality, kinds of arrows (epics, monics, isomorphisms …)
3. Functors
4. More about functors and categories (and the category of categories!)
5. Natural transformations (with rather more than usual on the motivation)
6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
7. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
8. An aside on Cayley’s Theorem
9. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
10. Representables (definitions, examples, universal elements, the category of elements).
11. First examples of limits (terminal objects, products, equalizers and their duals)
12. Limits and colimits defined (cones, limit cones: pullbacks and upshots)
13. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).

Coming soon: a further chapter on limits (on functors and the preservation of limits) and then a chapter on Galois connections as a nice gentle lead-in to the chapters on adjoints.

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### 2 Responses to Notes on Category Theory v.4

1. Andrei says:

Are you still planning on writing that book on ordinal-theoretic proof theory?

• Peter Smith says:

Well, to be honest, the book I was writing (try to be clever about constructive ordinals) rather fell apart. So that project is very much on the back burner at the moment — though maybe I’ll get back to writing a shorter, more conventional, book in due course. At the moment, I’ve got rather tangled up in category theory (which is more fun to be involved in as Cambridge is full of category theorists).