Here is an updated version of my on-going Notes on Category Theory, now over 150 pp. long. I have added three new chapters, at last getting round to the high point of any introduction to category theory, i.e. the discussion of adjunctions. Most people seem to just dive in, things can get a bit hairy rather quickly, and only later do they mention, more or less in passing, the simpler special case of Galois connections between posets (which transmute into adjunctions between poset categories). If there’s some novelty in the Notes at this point, it’s in doing things the other way around. We first have a couple of chapters on Galois connections — one defining and illustrating this simple idea, the other discussing a special case of interest to the logic-minded. Only then do we get round to generalizing in a rather natural way. We then find e.g. that two equivalent standard definitions of Galois connections generalize to two standard definitions of adjunctions (presented without that background, it isn’t at all so predicatable that the definitions of adjunctions should come to the same). I do think this way in to the material is pretty helpful: I’ll be interested, eventually, in knowing how readers find it.
So we this is what we now cover:
- Categories defined
- Duality, kinds of arrows (epics, monics, isomorphisms …)
- 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)
- 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).
- Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)
- [NEW] Galois connections (warming up for the general discussion of adjoint functors by looking at a special case, functions that form a Galois connection)
- [NEW] An aside for logicians, concerning a well-known example, the Galois connection between syntax and semantics.
- [NEW] Adjoints introduced. [Two different definitions of adjoint functors, generalizing two different definitions of Galois connections; some examples of adjunctions; a proof that the two definitions are equivalent.]
There will certainly be a few more chapters on adjoints. But don’t hold your breath, with a family holiday coming up and some other commitments. I haven’t decided yet whether eventually to add a chapter or two on monads (for monads seem a standard next topic to cover — e.g. in the last main segment of the Part III Tripos category theory course this year, the last chapter of Awodey’s book). Watch this space.