Notes on Category Theory, v.8

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!

5 thoughts on “Notes on Category Theory, v.8”

  1. Many thanks for diligently writing these notes. I haven’t quite gotten around to more than skimming them yet, but I fully intend to read them properly in time!

    As a small point, but a helpful on nonetheless, could you perhaps fix the PDF page numbering to match the graphical page numbering?

    Regarding the actual content, do you bother motivating each section or are some simply presented in their full abstract glory?

    1. On the minor point, the latest version of Adobe Reader DC seems to be intelligent about page numbering — so putting “17” in the page number box takes you to main body page 17 of the Notes (ignoring the vii pages of prelims).

      There’s quite a bit of motivational discussion, which explains the length of the Notes! — though some topics get more extended motivation than others (in this present, not particularly polished, version).

      1. Good to hear about the motivational discussion. Personally, that’s what I like most when learning something new.

        As for the page numbering: you’re quite right. It was/is just page 1 that is ambiguous, since the front cover and actual page 1 both have that number. You can however set the cover page to some text (or an upper case Roman numeral) to remove this ambiguity. If you wouldn’t mind for a future version, that would be great.

  2. It’s a minor detail, but a curious one: if I click on “And what comes next?” in the table of contents, I’m taken to the last page before that section, rather than to the page on which it begins. This happens both when I view it in Firefox and when I use Preview (both on a Mac).

    1. Thanks for spotting that. I’d forgotten that dratted bug/feature of hyperref which means you have to insert “\phantomsection” before starred TOC entries … Corrected in v.8a, coming soon!

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