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.
10 thoughts on “Notes on Category Theory v.6”
I do not understand your first example in “Preservation of Limits” (chapter 15), starting where you say that “Since the empty set is not a product in Set…” I must have missed something, because I would have expected the empty set to be the product of itself and anything else. Similarly, the “non empty case” would be the one where C, D(i) and D(j) are all non empty, these sets would all be sent to 1 by the functor F, and the preservation of limits would in this case amounts to 1=1×1 (up to isomorphism, of course)…
Very intersting work anyway, thank you for all that.
Ouch! Yes, that does seem a foul-up. Let me think about the best repair …. Thanks for spotting this embarrassment!
Thanks for the notes! I find them mostly easy to read. Some comments/questions/corrections:
– In 5.5, page 37, you write that:
“We can think of whiskering as a special case of the horizontal composition of natural transformation where one of the functors is the identity functor”.
But shouldn’t that be “one of the natural transformations is the identity natural transformation”, since that’s what you show next in the same paragraph?
– Sec. 6.1, page 41:
Proof of Theorem 22. “Since F is monic” -> “Since F f is monic”.
After Def. 27, example (7): that’s correct but a bit confusing — I got lost on the directions. It might have been helpful to name the continuous map $f$ and its underlying function $F’ f$.
More generally, who’s going to be the target for these notes? This question is premature and complicated, hence I split it off to a separate comment.
As a (mathematically-inclined) computer scientist, I had trouble with the details of dual vector spaces. I’m very sad to say I had my linear algebra ten years ago and forgot enough (and the proof I did read is made of easy calculations, but right now they don’t seem memorable ones: http://perso.uclouvain.be/jean-pierre.tignol/MAT1231.pdf). Funnily enough, this example appears easier to follow in Awodey’s book (IIRC), since he presents less detail.
Other computer scientists are going to have trouble with topology, since it’s not part of usual CS curricula AFAIK; I follow along mostly with topological intuition about rubber. I’d expect that many philosophers have less such background.
On the other hand, I can’t suggest to reduce prerequisites too much — Awodey’s right when he says that category theory can’t be developed in a vacuum. In fact, I enjoyed those examples, even though they were harder to follow.
Many thanks for letting me know about the needed corrections!
Who is going to be the target for these notes? That’s a very good question, Paolo!
At the moment, the notes are intended mainly for a very particular audience of one, namely myself! Though of course I hope that some others (e.g. students taking the Part III course this year) might find the notes — even in their current state — useful in nailing down some of the basics of category theory.
But yes, when I get round to rewriting the notes from the beginning (and I can now see various ways of doing things a lot better) it would be good to have a more definite conception of what background to take for granted and what needs more explanation.
Maybe the approach that Tom Leinster takes in his book is the best one: throw in lots of examples, but keep signalling that you don’t need to understand all of them: it is good enough that *some* examples link up to what is already tolerably familiar. Though, as you say, philosophers with little mathematical background are always going to be struggling a bit!
In definition 91, page 138, “Let L is the set of sentences” should be “Let L be the set of sentences”.
In clause (2) of that definition, “be” should be “is”. (What is the script letter being defined in that clause, btw? S?)
Page 152, below the diagrams, there seems to be something missing after “Transpose via the natural isomorphism that defines the”.
That’s all I’ve noticed so far.
I still have a bit of a “so what?” feeling about Galois connections, though. Yes, it’s an abstract framework or pattern that appears all over the place, but it doesn’t seem a very interesting one. In Lawvere’s ‘Adjointness in foundations’ paper, he talks of “giving definability theory a new significance outside the realm of axiomatic classes”. That sounds like it might be interesting, but I don’t know what it amounts to.
Many thanks for the corrections! I’ve also noticed that some of the diagrams in Ch. 18 don’t follow the previous convention of using double arrows to indicate natural transformations. I’ll repair in due course.
What about the interest of Galois Connections? As I was redoing this material from an earlier handout — and because of a computer disaster a few years ago I happened not to have the original LaTeX source so I had to plod through reconstructing from a PDF — I too began to get a bit worried by how intrinsically interesting it is! So for my purposes now the selling point is that they provide a toy class of adjunctions. But I’ll think a bit more about this before the next version!
1. I hope you’re feeling better.
2. A question out of ignorance: does your sequencing (as you outline above) reflect the historical development or did the ur-texts of category leap in with the synoptic version?
1. Yes, feeling a lot better thanks!
2. Well, the sequencing reflects how the Part III course went this year (and in Julia Goedecke’s excellent notes from when she was giving the Part III course). But there are other ways of proceeding. After the basics, Peter Johnstone’s course used to go Natural transformations and Yoneda (NYT) — Adjoints — Limits — (then Monads etc.). Tom Leinster’s book goes Adjoints — NYT — Limits — Adjoints again. There seems something to be said for doing Limits — NYT — Adjoints — Limits again. The trouble is that aspects of elementary category theory hang together tightly in ways that means that there is no single ideal linear ordering. (But that’s why it is an expository challenge that I’ll be returning to!)
As to the line of historical development, I’ve got a reading list and will report back. Historically natural transformations come first in the story, but after that I’m not so clear. My sense is that the development was slow and messy.