For those interested in category theory, this is just to say that the page here of links to online resources (at intro and mid-level) has been updated, broken links repaired, a couple of links added. Please spread the word, and also please let me know about recent additions to the available materials. Since right now I’m not working in this area, I could very well have missed things.
For those not interested in category theory, here’s the view from Garret Hostel Bridge walking through town on Easter Sunday …
Before I retired it was on my most frequent route to walk or cycle to the Faculty … and the view has been so familiar for half a century. But it still lifts the heart.
I am a computer scientist trying to catch up on the paradigm of functional programming and therefore I want to learn the basics of category theory. I have just started with your “gentle introduction” but your book seems to be the first one I might be content with for the following reason: You take the delicate relation between category and set theory seriously and you tell me so as a reader. A few weeks ago (before I came across your book) I’ve bought Awodey’s “Category Theory”. On p. 17 he says that the category of pointed sets is isomorphic to the coslice theory 1/Sets while he had defined an isomorphism on p. 12 as a special arrow between objects in a category. The only category introduced so far that has categories as objects was Cat, the category of small categories and functors between them. So the only reasonable thing he could have meant is that the said categories of pointed sets and 1/Sets are isomorphic as objects in Cat. However, both of these categories are not in Cat, since they are not small (right?). This sloppiness drives me crazy. I am feeling as not taken serious as a reader. So, after having discovered your book I looked up what you have to say about the same topic: On p. 35 you say “Hence 1/Set is (or at least, in some strong sense to be later explained, comes to the same as) the category Set˚ of pointed sets.”. What a relief! I fully appreciate your pernicekty and honesty, especially as I always see Russel’s paradox lurking behind the next corner and as I am interested in (besides the theoretical base of the functional programming paradigm) in the relation between category theory and set theory. My impression from the first pages of your book is that you (contrary to Awodey) are not trying to sweep aside my qualms about potential paradoxes/inconsistencies with an easy hand nor are you trying to speculate on my forgetfulness about the precise content of definitions a few pages ago. You book seems to be made for people like me. Thank you very much for it.
Thanks for the friendly comments! I hope to get back to thinking about category theory, for better or worse, once I’ve got my current book project off to the press — so it is always good to hear that someone found my notes helpful. I’d like to get a lot clearer in my own mind about the role a background set theory plays in category theory.
In the meantime I am at chapter 10 of your book and unfortunately my misgivings about potential Russel-like paradoxes have returned: I am distrustful about the concept of a cone (Def. 54, p. 84): Your notation [C,c_j] pretends that entity to be a pair while really j is understood as an index ranging over an appropriate set(?)/(meta-theoretical) collection(?)/family(?) such that every diagram object is indexed. So, w.r.t. size considerations a cone is the same thing as a category, it consists of objects and arrows. And we have already learnt that only for small categories we have a definitely safe implementation procedure, viz. sets, rendering categories well-defined mathematical objects such that they can, e.g., serve themselves as objects (in the category-theoretical sense) within another category, viz. Cat, the category of small categories.
So, it seems to me, a cone cannot be defined as you did it without risking to run into Russel-like paradoxes. I was first skipping over my qualms but now as I am at Theorem 50, p. 89 where you consider an (unrestricted) category itself a diagram, the problem is returning with full urgency. So what is at the heart of the problem (according to my layman considerations as a learner)? It seems to me that without restricting to small categories as base ones you have no guarantee that you can tell apart two potentially different cones. You have no guarantee that you have or find a procedure to decide whether two “cones” are the same or not, as you have no guaranteed procedure to tell two (unrestricted) categories apart. This is the reason or at least one way to state the problem (as I have understood it) that prevent us to immediately define a category CAT of all categories so that we have to be satisfied with Cat, the category of all small categories as the next best thing. The problem here becomes acute within Def. 57, p. 87 where you officially ennoble cones to (category-theoretical) objects in a respective cone category.
Awodey, as expected, generously skips over this problem alltogether (what was the reason why I got frustrated in the first place and picked up your book instead, as I have told you before): When defining a cone on p. 101 Awodey says “a cone consists of an object […] and a family of arrows […]”. Ah, yes. A “family”. What the hell is a family?
My question to you is? Will anything essential of category theory be lost later on if cones are restricted to small base categories? If so, that seems to me (again as a layman and learner) as a serious flaw in this mathematical discipline itself. Otherwise, I suggest to change the definitions and to restrict cones to small base categories. What do you think? Thank you very much in advance.
I keep looking for something that would get me past these two reactions:
* When I look at a ‘theoretical’ category theory text, a great feeling of “so what?” comes over me. There seems to be an awful lot of fiddly terminology with nothing that makes it seem it would be worth the trouble to learn it.
* When I look at an applied text, much of it seems an exercise in propaganda, trying to make it seem category theory is necessary when it isn’t, and making ideas harder to undertand by recasting them in excessively abstract category theoretical terms.
I had hopes for the new Fong and Spikak Seven Sketches book (an Invitation to Applied Category Throey), but it seems to have exactly that problem. Look at the “prepare lemon meringue pie” diagram on page iv (and again on page 40). On p iv, it claims
The basic idea being communicated by the diagram is indeed easy to understand, but it’s tendentious, at best, to claim the basic idea that people understand is monoidal categories. And of course it could be formalised, but monmoidal categories are not the only way to do that, and I doubt it’s the best one.
Even much of the maths used in — such as graphs, preorders, and monotone maps — isn’t category theory and doesn’t need a category theoretical setting. Yet the book tries to make them seem more category theoretic, for instance by defining ‘graph’ as what’s basically a directed graph but with the edges called “arrows”.
I’m not at all unsympathetic to your reaction. Indeed my own amateur philosopher’s interest is in trying to think a bit about what’s hype and what’s real content. And I agree that in some intro presentations there’s too much that looks like that the first for my austere tastes!