At long last — more or less exactly nine months since I started intermittently writing them — there is a first complete version of my Notes on Category theory (as they are now called). Or at least, the Notes are complete in the sense that I don’t intend to press on to add further chapters on significant new topics like monads or abelian categories. Rather the current plan is to leave these notes in more or less their present form, for all their shortcomings (and I hope in due course to start writing a differently organized, more discursive, bigger and better version). Still, I would very much like to hear about errors of one kind or another. And I’ll no doubt issue occasional “maintenance upgrades” when I hear about mistakes or spot passages which really won’t do — and perhaps I might add more illustrative examples or even new sections here or there to round out the treatment of existing topics where the coverage in retrospect seems too skimpy.
Since the previous version, I have expanded the chapter on some general results about adjunction, and added a chapter on adjunctions and limits. This has entailed quite a bit of going back to earlier chapters, adding material to smooth the route to later theorems. I finish up by waving my hands at, though not elucidating the content of, the Adjoint Functor Theorems, General and Special. But it is a non-trivial expositional task to explain these (the technical proofs aren’t hard; what isn’t so easy is to see is the motivation for the various new concepts — like the ‘solution set condition’ — which they involve). I’m not sure I yet have a sufficiently good grip on the place of these theorems in the scheme of things to give an illuminating account of the motivations. So I’m at the moment shirking the task of trying to explain more.
But in any case, the Adjoint Functor Theorems arguably sit on one of the boundaries between basic category theory and the beginnings of more serious stuff. So given the intended limited remit of the Notes (now highlighted by calling them notes on Basic Category Theory), the Theorems mark a reasonable point at which to stop for now.
So, with that by way of preamble, here is the new version of the Notes (190 pages). [Link updated to version 8a] Enjoy!