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