It’s a balancing act. On the one hand, I don’t want to annoy readers with over-frequent announcements of minor revisions. On the other hand, I don’t want to keep propagating flawed versions when I have an improved offering in hand!
Anyway, I’ve been reading through the first 11 chapters of the Notes making some minor corrections and other changes. I’ve also had some much appreciated corrections of a few mistakes in later chapters from Alessandro Stecchina. Since I think there will be something of a pause before I can press on to re-read the rest of the Notes, here’s an interim update, to
version 8a version 8b of the Notes.
5 thoughts on “Notes on Basic Category Theory, v.8a/8b”
Seeing this post about your notes has led me to consider again a question that’s been bugging me for a while now: why haven’t I already learned more category theory? I have your notes, Leinster’s book, some other books. I’m trying to commit myself enough that I’ll feel I should carry on, but some things are putting me off. One is a “so what?” feeling. Yes, there will be some high level structures and patterns, just as there are in, say, group theory. But why should I care? What’s actually interesting about them, beyond being at a higher level than groups? The other is that category theory, or what at least what some people say about it, seems excessively ideological, which both provides and then undercuts another motive for tackling the subject.
These issues usually arise quite quickly when I start reading about category theory. For instance, when you answer “What do we gain by going up another level of abstraction or two?” on page 1, you quote Tom Leinster about the bird’s eye view. That makes sense to me. That sounds good. As does the prospect of ” large-scale but still illuminating patterns in mathematics which aren’t otherwise discernible.” But then, for “a glimpse ahead” and “the very briskest of hints about what some of these patterns might turn out to be” — only a glimpse, only brisk hints — we’re referred to e.g. The Joy of Cats. So it doesn’t look like there will be much about those patterns in your notes or even in the book you refer to. I’m left with the prospect of learning some only moderately interesting maths … to what end?
Then there’s the ideological part. On page three, you write “the central idea of category theory is that we should probe objects by considering the morphisms between them, not that we should write objects out of the story”; but getting rid of objects — or at least of any ontological commitment to them — does seem to be one of the central ideas, at least in what some advocates of category theory write.
Your first example of a category is Set, and I think you’re right to raise the issue of what exactly that category is. I don’t think it quite works to say the sets in Set can be whatever you’re used to thinking of as sets. They can’t be NF sets, for a start; and I’m not sure they can be ZFC sets either. (If they’re equivalent to the cumulative hierarchy up to an inaccessible, for example, or if they include urelements, then they’re not quite sets as in NFC; they’re some variation or restricted form.) One of my reasons for wanting to learn more category theory is to get a better understanding of such issues.
But that means dealing with the ideology, and I find it both interesting and a bit … annoying.
There is a pervasive undercurrent of hostility to set theory in some of the category theory literature (at least if set theory means something like ZFC). There’s even a glimpse of it in your notes on p 65 where you say: “Not perhaps if we have ambitions for category theory as a more democratic way of organizing the mathematical universe, which provides an alternative to set-theoretic imperialism.”
About ideology, it’s true that often category theorists can be rather tiresome and supercilious about the rest of us poor schmucks (who they’re apt to lump together).
It must be admitted though, that pretty well anyone in academia will find in their more honest memories occasions when their toes have curled on behalf of their younger selves.
I could go on in that vein, but I wanted to mention something that first really impressed me about category theory, namely the Yoneda lemma, by which I mean the (natural!) isomorphism between Set^C[C[A,_],F] and F(A) when C is locally small (like Set) and F : C -> Set. (I hope I’ve got that right; may have dropped an op somewhere.) On the left we have natural transformations between functors (of a special form) and arbitrary functors. Just looking at the definition of natural transformation, the collection of those things might be horrifically large! But it’s isomorphic to a *set*!! Isn’t that amazing? One can push this a little further beyond the highly special form C[A,_] to set-indexed coproducts of such functors. Quite apart from being rather surprising, allow me to say that this has a lot of mileage in real (technical) life.
As a non-category-theorist who is nevertheless rather convinced of the value of asking myself category-theoretic questions about the gadgets I am trying to think about, I once (preposterously) resolved to keep a list of such things, which (despite my background and general curmudgeonality(?)), I’d be prepared to say on behalf of the category-theoretic perspective. It’s a bit hazy beyond the first entry (above), but there have been several occasions when I’ve been rather delighted by the clarity that
category theory can bring (at a low level) to the role of, I don’t know, maybe distributivity of this over that, in some argument. (For me these tend to be occasions when I am teasing apart the synapses, as it were, of an argument that just slithered out of my intuition.)
I’d love let fly about the pretentiousness of category theory with respect to “foundations”. (At least they call the concept into well-deserved question.) I’ll repress that however, and just refer to the writings of Sol Feferman on the topic, many of which can be found at https://math.stanford.edu/~feferman/papers.html .
When, many decades ago, I first listened to category theorists ranting about the preposterousness of set theory, and its oh-so-laughable binary relation of set-membership, I remember distinctly thinking: well you have a binary symbol here which is just as mysterious, namely the symbol for morphism-equality. I have to admit, in view of recent developments in homotopy type theory, that it’s hard to get rid of a certain supercilious smugness for having that thought. My toes will curl in hell, I expect.
[To P.H.] Yes I’m rather inclined to be in the Feferman camp here. Though still labouring away to get to the point where I have a confident enough grip on what category theorists are up to endorse his kind of critique of overblown claims.
As to HoTT, my copy of the book remains still mostly unread and my grip on what the heck the cool kids are up entirely tenuous.
[To R.M.] In fact, I do rather share your skepticism about the wilder claims of some over-enthusiastic proponents of category theory. And my plan all along has been to get my head round enough category theory to be in a position to do some philosophical deflation (not of the maths of course but of some of the surrounding chat).
Some years ago, I had a similar experience with chaos theory, about which some people made all kinds of of wild claims. I had fun reading up a lot of applied maths, and then wrote Explaining Chaos which explains enough of the maths to make clear what’s going on and so deflate the wilder claims. Well, with category theory I’m still at the stage of trying to understand enough of the maths, which — sad to relate — I’m finding a fun work-out for my ageing grey cells. The Notes just show me grappling with the some of the basics. But yes, I predict that the more I understand, the more skeptical I will tend to get about some of the wilder claims made by some enthusiasts.
As to some specific comments, (1) I only mean the “glimpse ahead” to be for those who couldn’t wait to see how things pan out in the Notes and beyond — but yes, I must come back to say more about these patterns glimpsed from on high.
(2) You say it won’t do to say that the sets in Set can be whatever you’re used to thinking of as sets, and suggest e.g. that they can’t be NF sets. Well yes, eventually, if we want to assume Set is Cartesian closed, that rules out it being the category of NF sets. But at the outset we can stay more relaxed. It is a nice question — which I agree in the current version doesn’t ever get addressed — of just when in the game we start making assumptions about Set that force our hand.
(3) I didn’t mean to sign up to any kind of hostility to set theory, which is lovely in its place. I’m only hostile to over-quick, knee-jerk, claims about set theory being the one true foundation for modern mathematics, whatever that might mean.
I didn’t mean to suggest you were hostile to set theory yourself; nonetheless, some of the hostility seems to have seeped in (likening set theory to imperialism, and describing category theory as more democratic), although in a light-hearted form.
That’s not inevitable when category theory is discussed, but it is quite common. I looked at some category theory books today, trying to get a better feel for some of the issues. Simmons’s Introduction was refreshingly free of ‘ideology’, as was Lawvere and Schanuel’s Conceptual Mathematics (which surprised me, though I may have missed something).
However, Leinster’s Chapter 3, ‘Interlude on Sets’, is quite ideological, in the usual hostile-to-set-theory way, especially in the ‘Historical Remarks’; and as usually happens, it makes a number of tendentious (or at least contestable) philosophical moves and claims. Even what is says about types is, I think, misleading. (The ‘inbuilt sense of type’ which he uses to introduce the idea is very different from the types in programming languages that he then moves to, because in the ‘inbuilt sense’, it’s the objects that have types, not the words (variables and expressions) used to refer to the objects. Indeed, the more ideological sort of programming language type theorist is hostile to the languages in which it’s the objects that have types.)
I also, while looking around online, ran across an interesting interview with Lawvere which I wanted to mention because of something it says about Godel’s incompleteness theorems:
The same comment appears in the introduction to a reprint of that article, but without the footnote that explains the part about the ‘agenda’ becoming explicit:
The rhetoric is interesting — demystify, very simple algebra, declared, suspicion among scientists, movements, agenda, organized attempts — and though the paragraph refers to Godel’s “great mathematical work”, and may have a point (to an extent) about how incompleteness is sometimes used, it also seems to be trying to give the impression that much too much is made of what category theory lets us see is really just some “very simple algebra”.