Dear all, Glasgow is just now introducing a Masters-level mathematics programme, and I'm teaching the Category Theory course. I'm looking for suggestions on a particular aspect of teaching it. It's a question of "size". Most of the times I've taught category theory previously were at Cambridge, where the students are exposed to ZFC-style set theory as undergraduates. Every year there'd be a few people who'd really worry about the set-theoretic validity of category theory: "doesn't Russell's paradox forbid a category of sets?", etc. I'd tell them, essentially, not to think about it; one can make a distinction between "small" and "large" collections, and experience shows that this suffices. Not a profound answer, but there you are. At Glasgow I'm going to have the opposite problem. Undergraduates here do no set theory of any kind. So, for instance, there's no reason why they should have heard of ZFC, or that there are collections "too big to be sets". Be careful what you wish for: after years of telling Cambridge students to forget their set theory, I now have students with no set theory to forget. And the question I'm having trouble answering is this: what do I need to tell them about sets? I can't tell them nothing, as far as I can see. For instance, I want them to know that the category of groups has "all" limits; but of course, Grp doesn't really have all limits, only small limits, so they'll need to know what "small" means. Later, I'll want to teach the Adjoint Functor Theorems. A rough and ready solution would be to tell them that there is a distinction between "small" and "large" collections, otherwise known as "sets" and "proper classes". This would necessitate giving them an example of a large collection, and I guess the obvious choice is the class in Russell's Paradox. But then I'd have to tell them that this is exactly the kind of thing that they shouldn't be thinking about! It's hardly satisfactory. There's probably a better solution involving an axiomatization of the category of sets (along the lines of the Lawvere-Rosebrugh book), or at least a listing of some its properties. I have two difficulties here. One - which readers of the list may be able to help me with - is that I haven't figured out how this would work in practice: for instance, how it would feed into the statement above on the completeness of Grp. Does anyone have experience of this? The other is that I haven't got room to be too radical, as the syllabus is already set (categories, functors, transformations; adjunctions, representables, limits; monads and/or monoidal categories). In a way this is an ideal situation: a classful of minds innocent of ZFC, able to come at set theory in a completely fresh way. I'd very much appreciate suggestions on how best to use this freedom. Tom
A rough and ready solution would be to tell them that there is a distinction between "small" and "large" collections, otherwise known as "sets" and "proper classes". This would necessitate giving them an example of a large collection, and I guess the obvious choice is the class in Russell's Paradox. [...]
There's probably a better solution involving an axiomatization of the category of sets (along the lines of the Lawvere-Rosebrugh book), or at least a listing of some its properties.
The question is more when than whether to bring up this distinction. Although the Lawvere-Rosebrugh book doesn't define "large" it does define "small relative to Set" in the antepenultimate paragraph of the book (p.250), namely as "can be parameterized by an object of Set". Near the midpoint of the book (p.130) is the statement of Cantor's theorem X < 2^X with the consequence that Set cannot be parameterized by any of its objects (and hence by the definition on p.250 cannot be small relative to itself). In contrast Borceux in his 3-volume series gets the size issue out of the way on pages 1-4 of Volume 1. CTWM is in between, addressing it on p.22 after covering categories, natural transformations, monics, epis, and zeros. One benefit of getting the distinction out of the way near the beginning is that the students won't feel so mystified when they run across it while reading other category-relevant material (as the better students will). The combinatorics of sets (n^m functions from a set of m elements to a set of n elements, etc., which they definitely should know) in no way prepares one for the possibility of an object larger than any set, for which Cantor's theorem is very helpful. Of the above three positionings, I like CTWM's best: early on, yet not so early as to exaggerate its importance relative to the fundamental concepts of CT. Not to say that CTWM starts out ideally. Spending three pages defining "metacategory" and then defining "category" as "any interpretation of the category axioms within set theory" is impenetrably idiosyncratic for students used to more conventional introductions in their other pure maths courses. Once past the size issue Borceux is much more conventional and direct, except for the relatively mild criticism that his definition of "category" is actually the definition of "locally small category." But that's not at all the stumbling block to understanding that "metacategory" presents, in fact if anything it is helpful not to be distracted at the outset by the prospect of large homobjects. While on the topic of size of homobjects, what drawbacks are there to regarding both the objects and homobjects of any n-category as all lying within the n-th Grothendieck universe for a suitable hierarchy U_1 < U_2 < ... of such (with U_0 = 1)? Although admittedly idiosyncratic, it seems very natural to account for large homobjects in a category C with the explanation that C is really a 2-category, whether or not one is making use of the 2-cells. I can see a methodological objection, namely that there is no logical connection between size and existence of n-cells for a given n. Does it create any actual difficulty somewhere? Vaughan
--- Jeff Egger <jeffegger@yahoo.ca> wrote:
Date: Thu, 30 Aug 2007 13:48:36 -0400 (EDT) From: Jeff Egger <jeffegger@yahoo.ca> Subject: Re: categories: Teaching Category Theory To: Tom Leinster <t.leinster@maths.gla.ac.uk>
Dear Tom,
I find that the set/class distinction is much less compelling than the type/collection distinction, so my initial reaction is that one should develop a kind of "naive type theory" to replace "naive set theory" ---but I don't know to what extent it is possible to do this in a pedagogically sound fashion.
[Googling "naive type theory" yields some interesting-looking articles, but I haven't really had time to look at any of them in anything approaching a serious fashion.]
The central tenet of NTT should be the intuition that one can't compare apples and oranges. In particular, you can't ask whether two things are equal unless they were already "of the same type", which is to say that they were chosen from the same set to begin with.
[Interestingly, there exist better motivating examples than "apples and oranges". Does the speed of light equal the charge of a positron, for example? Of course one could say that the answer is yes if we measure the speeds in light-seconds per second and charges in elementary charges, or we could say that the answer is no if we use more conventional units such as km/h and coulombs. But if we try to conceptualise physical quantities as real entities independent of a choice of unit of measurement, then we recognise the question itself as flawed.]
Of course, elementhood should also not be a global predicate, for otherwise we could subvert the non-existence of a global equality predicate by asserting two things to be equal if they are equal in every type to which they both belong.
The non-existence of a global elementhood predicate renders the extensionality axiom of conventional set theory meaningless. This, in turn, calls into question whether equality of types is a meaningful predicate. But the existence of a type of types would entail the existence of such a predicate, and thus we are led to a situation where the non-existence of a type of all types can be regarded as a feature, not as a bug.
You see what I really have in mind is not so much topos theory (which you might have suspected at first), but FOLDS. [But NTT should be set up in such a way that elementary topos theory becomes a (or even, the) natural result of attempting to formalise one's naive intuitions about types. For example, one can talk about (product- and power-)type constructors in a naive way...I think. Ideally, I would hope that naturality could be adequately described in terms of polymorphic lambda-calculus---but even I wouldn't suggest springing that on an unsuspecting first-term graduate student.]
Here is another helpful intuition for students: a set/class/type/ collection should not be thought of as a "glass box", but rather as a black box with a button: when you push the button it gives you, not an element of the set, but a little receipt bearing the name of an element of the set. [Riders of the Montreal metro system may recognise the boxes from which one obtains bus transfers, which held a strange fascination over me when I was a child.] For arbitrary collections, it is possible that an element have more than one name---and hence, when you ask for two elements, you may in fact receive two names of the same element, _and_ be left none the wiser for it. A type is (naively) a collection for which every element has a unique name.
Now using NTT/FOLDS as a basis for category theory does restrict one to dealing with locally small categories (if, one regards types as necessarily "smaller" than arbitrary collections---which is not as easily justifiable as it might seem), but I would argue that's not such a great loss. [In my experience, non-category-theorists, when asked to provide a definition of category, almost uniformly supply (what amounts to) the definition of an enriched category, in the case V=Set---which I find quite intriguing.] It also destroys the notion of skeletal category, which is probably a good thing too.
I hope this helps---I was originally planning to write a lot more (and might still do so).
Cheers, Jeff.
Dear Tom, Here is a rationale for a "cop-out" answer. When I taught categories at Imperial I had 10 lectures in a Computer Science department, so there was a limit to what I could do. For the adjoint functor theorem, I just taught the easy direction - that a (let's say) left adjoint preserves all colimits that exist. That's already an extremely useful thing to know. For the converse (a functor from a cocomplete category is a left adjoint if ...), one can already prove it for posets as a special case. I didn't go any further because (a) I didn't have time, and (b) - like you - I didn't know how to explain the set/class issues. But in retrospect I'm completely happy with that because the result relies on classical reasoning principles. In practice there is a lot to be said for constructing a right adjoint explicitly by other means. Checking that colimits are preserved then becomes a prudent check before you spend time looking for an adjoint. You also mention the issue of the category of groups having "all" limits. It seems to me that if you show them how infinite products have the universal property, and how then other limits can be constructed, then you'll have shown them the main thing they need to know. I would keep the set/class issue as a secret aside for those who have seen it before. Regards, Steve Vickers. On 27 Aug 2007, at 02:58, Tom Leinster wrote:
Dear all,
Glasgow is just now introducing a Masters-level mathematics programme, and I'm teaching the Category Theory course. I'm looking for suggestions on a particular aspect of teaching it.
It's a question of "size". Most of the times I've taught category theory previously were at Cambridge, where the students are exposed to ZFC- style set theory as undergraduates. Every year there'd be a few people who'd really worry about the set-theoretic validity of category theory: "doesn't Russell's paradox forbid a category of sets?", etc. I'd tell them, essentially, not to think about it; one can make a distinction between "small" and "large" collections, and experience shows that this suffices. Not a profound answer, but there you are.
At Glasgow I'm going to have the opposite problem. Undergraduates here do no set theory of any kind. So, for instance, there's no reason why they should have heard of ZFC, or that there are collections "too big to be sets". Be careful what you wish for: after years of telling Cambridge students to forget their set theory, I now have students with no set theory to forget. And the question I'm having trouble answering is this: what do I need to tell them about sets?
I can't tell them nothing, as far as I can see. For instance, I want them to know that the category of groups has "all" limits; but of course, Grp doesn't really have all limits, only small limits, so they'll need to know what "small" means. Later, I'll want to teach the Adjoint Functor Theorems.
A rough and ready solution would be to tell them that there is a distinction between "small" and "large" collections, otherwise known as "sets" and "proper classes". This would necessitate giving them an example of a large collection, and I guess the obvious choice is the class in Russell's Paradox. But then I'd have to tell them that this is exactly the kind of thing that they shouldn't be thinking about! It's hardly satisfactory.
There's probably a better solution involving an axiomatization of the category of sets (along the lines of the Lawvere-Rosebrugh book), or at least a listing of some its properties. I have two difficulties here. One - which readers of the list may be able to help me with - is that I haven't figured out how this would work in practice: for instance, how it would feed into the statement above on the completeness of Grp. Does anyone have experience of this? The other is that I haven't got room to be too radical, as the syllabus is already set (categories, functors, transformations; adjunctions, representables, limits; monads and/or monoidal categories).
In a way this is an ideal situation: a classful of minds innocent of ZFC, able to come at set theory in a completely fresh way. I'd very much appreciate suggestions on how best to use this freedom.
Tom
participants (4)
-
Jeff Egger -
Steve Vickers -
Tom Leinster -
Vaughan Pratt