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