Teaching Category Theory

["Tom Leinster" <t.leinster@maths.gla.ac.uk>]

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