Re: Teaching Category Theory

Re: Teaching Category Theory

[Jeff Egger]

--- 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.
> 
> 
> 

Re: Teaching Category Theory

[Ronnie Brown]

One would like to leave students with a very positive attitude. The
following quotation , from the Stanford Encyclopedia of Philosophy,  might
help:
" Category theory has come to occupy a central position in contemporary
mathematics and theoretical computer science, and is also applied to
mathematical physics. ......."
http://plato.stanford.edu/entries/category-theory/

I am writing as someone who has come into category theory from algebraic
topology, and been struck by the utility of the language and results for
what I needed over the years (and by the welcome).

A separate matter, but intriguingly related,  is categories and groupoids as
sources of useful algebraic structures. This seems connected with the notion
of partial operations, and so my notion of `higher dimensional group theory'
and `higher dimensional algebra' is that of  studying algebraic structures
with partial operations defined under geometric conditions.  This concurs
with the vision in
Higgins, Philip J. Algebras with a scheme of operators.  Math. Nachr. 27
1963 115--132.
In this view, the objects of a category play a key role.
This has covered the developments I had in mind for modelling some
underlying structures in homotopy theory, and which were developed with
Philip Higgins, and later with Loday. Relevant was Philip's reporting of the
view of Philip Hall that one should study the algebra that arises naturally
from the geometry without trying to force the algebra  into a preconceived
mould.

In the late 1960s, when Bill Cockcroft and I received notes from Saunders of
lectures on category theory for our comments, Bill and I replied that what
we really wanted was `Categories for the working mathematician'. I still
hold to that. To me this means general theory with specific examples which
show how the general theory makes life easier, even controls the
calculations. Eilenberg insisted a construction should be defined, and its
properties developed, in terms of the universal property, which should also
explain existence.

So when dealing with structures at various levels it is very useful to know
left adjoints commute with colimits, right adjoints commute with limits, and
this can tell one how to compute colimits and limits. This also leads to
induced constructions (change of base).

I have  recently found  uses (to me!)  of fibrations of categories: the
inclusion of a fibre preserves connected colimits. Simple examples of the
use of this are:
Ob: Groupoids \to Sets,
forget: (groupoid modules) \to groupoids;
forget: (2-Cat) \to Cat
and compositions of these.
Of course it was generalisations of the van Kampen theorem to higher
dimensions, and the (previously rare) use in homotopy theory of colimits of
algebraic structures,  that made it useful to do such computations.

I have only recently really understood the notion of dense subcategory, and
its use for representing an object as a coend.

What I have not done is use the theory of monads. Is this ignorance on my
part? I am happy to be enlightened!

One of the points of a course for the students might be `need to know'.
Hence the need for explicit  and varied examples. How to balance this with
theory?

Ronnie
www.bangor.ac.uk/r.brown

----- Original Message -----
From: "Jeff Egger" <jeffegger@yahoo.ca>
To: <categories@mta.ca>
Sent: Tuesday, September 04, 2007 5:30 PM
Subject: categories: Re: Teaching Category Theory


--- Michael Shulman <shulman@math.uchicago.edu> wrote:

> My guess would be that it's because for non-category theorists, many
> (perhaps most) categories which arise in practice are enriched (over
> something more exotic than Set), while few are internal (to something
> more exotic than Set).

I'm not sure I agree with that: internal groupoids, at the very least,
show up in a variety of situations which non-category theorists can be,
and are, interested in.  Perhaps one of the reasons why some people try
to deal with groupoids as if they weren't a special case of categories
is because they never thought of categories in any other way than as a
mass of hom-sets.

> Even when working over Set, I think it's fair
> to say that the vast majority of categories arising in mathematical
> practice are locally small.

Now I do think there is a good reason for this, which is the fact
that in functorial semantics (by which I don't just mean the original,
universal-algebraic, case), the domain category is typically small.
Raising to a small power does not destroy local smallness.

> Since in general, neither enriched nor internal category theory is a
> special case of the other, it doesn't seem justified to me to consider
> either one as "more primitive".

I agree with this entirely, of course.  It follows that, in a first
course on category theory, one should present both styles of definition
as soon as possible.  This, in turn, suggests (but does not prove) that
one should not sweep size distinctions under the carpet.

> Actually, currently my favorite level of generality is something I
> call a "monoidal fibration".  Roughly, the idea is that you have two
> different "base" categories, S and V, such that the object-of-objects
> comes from S while the object-of-morphisms comes from V.  When S and V
> are the same, you get internal categories, and when S=Set, you get
> classical enriched categories.  This could be regarded as "explaining"
> the coincidence of internal and enriched categories for V=Set.  I
> wrote a bit about this at the end of "Framed Bicategories and Monoidal
> Fibrations" (arXiv:0706.1286), but I intend to say more in a
> forthcoming paper.

I look forward to it!

Cheers,
Jeff.




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Re: Teaching Category Theory

[Michal Przybylek]

"Jeff Egger" <jeffegger@yahoo.ca> wrote:

> --- Michael Shulman <shulman@math.uchicago.edu> wrote:

> My guess would be that it's because for non-category theorists, many
> (perhaps most) categories which arise in practice are enriched (over
> something more exotic than Set), while few are internal (to something
> more exotic than Set).

I think it's not that.

It's just because the concept of internal category is in some sense subsumed
by the concept of fibration. And in practical situations we prefer to work
with fibrations rather than internal categories.

> Actually, currently my favorite level of generality is something I
> call a "monoidal fibration".  Roughly, the idea is that you have two
> different "base" categories, S and V, such that the object-of-objects
> comes from S while the object-of-morphisms comes from V.  When S and V
> are the same, you get internal categories, and when S=Set, you get
> classical enriched categories.

What do you mean by "you get internal/enriched categories" ? Do you have a
2-equivalence between the 2-category of all S-internal (resp. enriched)
categories (S-internal functors, S-internal natural transformations) and the
2-category of your categories ? I'm asking because I have encountered some
difficulties here (i.e. in my framework some diagrams are not willing to
commute "on the nose").


Best regards,
Michal R. Przybylek

Re: Teaching Category Theory

[Michal Przybylek]

"Jeff Egger" <jeffegger@yahoo.ca> wrote:

> --- Michael Shulman <shulman@math.uchicago.edu> wrote:

[I don't see the original post, so I'm responding here]

> Actually, currently my favorite level of generality is something I
> call a "monoidal fibration".  Roughly, the idea is that you have two
> different "base" categories, S and V, such that the object-of-objects
> comes from S while the object-of-morphisms comes from V.  When S and V
> are the same, you get internal categories, and when S=Set, you get
> classical enriched categories.  This could be regarded as "explaining"
> the coincidence of internal and enriched categories for V=Set.

Heh... I'm studying the same problem as a part of my Master Thesis (under
supervision of prof. Andrzej Tarlecki), but fortunately :-) in a bit
different framework. The chief concept of my work is a definition of a
category ("elementary category") in a fibred monoidal category (i.e. each
fibre is monoidal and reindexing functors preserve the monoidal structure)
over a base category with binary products. Than, roughly speaking, for a
category C with finite limits, C-enriched categories are just "Fam :
Fam(C) -> Set"-elementary categories, and C-internal categories are just
"Cod : C^{->} -> C"-elementary categories. It turns out (if I didn't make
mistakes :-)), that when C has Set-indexed coproducts, than there is an
adjunction between the global section functor C(1, -) : Cod -> Fam and the
"coproduct functor" \coprod_{-}(1) : Fam -> Cod. Furthermore, if the
coproducts are universal, than these functors are fibred and preserves the
monoidal structures, and if additionally all global sections in C are
disjoint (i.e. the pullback of two different global section is an initial
object) than this adjunction is an equivalence of categories (these results
give us approximations C-internal categories ---> C-enriched categories and
in the other direction).


Best regards,
Michal R. Przybylek

Re: Teaching Category Theory

[Jeff Egger]

--- Michael Shulman <shulman@math.uchicago.edu> wrote:

> My guess would be that it's because for non-category theorists, many
> (perhaps most) categories which arise in practice are enriched (over
> something more exotic than Set), while few are internal (to something
> more exotic than Set). 

I'm not sure I agree with that: internal groupoids, at the very least,
show up in a variety of situations which non-category theorists can be,
and are, interested in.  Perhaps one of the reasons why some people try
to deal with groupoids as if they weren't a special case of categories
is because they never thought of categories in any other way than as a 
mass of hom-sets.

> Even when working over Set, I think it's fair
> to say that the vast majority of categories arising in mathematical
> practice are locally small.

Now I do think there is a good reason for this, which is the fact 
that in functorial semantics (by which I don't just mean the original, 
universal-algebraic, case), the domain category is typically small.
Raising to a small power does not destroy local smallness.

> Since in general, neither enriched nor internal category theory is a
> special case of the other, it doesn't seem justified to me to consider
> either one as "more primitive". 

I agree with this entirely, of course.  It follows that, in a first 
course on category theory, one should present both styles of definition 
as soon as possible.  This, in turn, suggests (but does not prove) that 
one should not sweep size distinctions under the carpet.  

> Actually, currently my favorite level of generality is something I
> call a "monoidal fibration".  Roughly, the idea is that you have two
> different "base" categories, S and V, such that the object-of-objects
> comes from S while the object-of-morphisms comes from V.  When S and V
> are the same, you get internal categories, and when S=Set, you get
> classical enriched categories.  This could be regarded as "explaining"
> the coincidence of internal and enriched categories for V=Set.  I
> wrote a bit about this at the end of "Framed Bicategories and Monoidal
> Fibrations" (arXiv:0706.1286), but I intend to say more in a
> forthcoming paper.

I look forward to it!

Cheers,
Jeff.

Re: Teaching Category Theory

[Michael Shulman]

[Note from moderator: a response to this item from Jeff Egger was posted
earlier, but the original was not... ]

On 8/31/07, Jeff Egger <jeffegger@yahoo.ca> wrote:
> Actually, what I find intriguing is that it is the definition of
> enriched category which seems to have priority over the definition
> of internal category.  There are, I suppose, historical reasons for
> this (pre-1960 the focus tended to be on AbGp-enriched categories)
> ---but I think it fair to say that (for as long as I can remember,
> which obviously isn't that long from a "historical" perspective)
> the majority of category theorists tend to adopt the internal
> category style of definition (of category) as more primitive.

My guess would be that it's because for non-category theorists, many
(perhaps most) categories which arise in practice are enriched (over
something more exotic than Set), while few are internal (to something
more exotic than Set).  Even when working over Set, I think it's fair
to say that the vast majority of categories arising in mathematical
practice are locally small.

Since in general, neither enriched nor internal category theory is a
special case of the other, it doesn't seem justified to me to consider
either one as "more primitive".  However, it's worth pointing out that
both are a special case of categories enriched in a bicategory, or in
a double category.

Actually, currently my favorite level of generality is something I
call a "monoidal fibration".  Roughly, the idea is that you have two
different "base" categories, S and V, such that the object-of-objects
comes from S while the object-of-morphisms comes from V.  When S and V
are the same, you get internal categories, and when S=Set, you get
classical enriched categories.  This could be regarded as "explaining"
the coincidence of internal and enriched categories for V=Set.  I
wrote a bit about this at the end of "Framed Bicategories and Monoidal
Fibrations" (arXiv:0706.1286), but I intend to say more in a
forthcoming paper.

Mike

Re: Teaching Category Theory

[Jeff Egger]

--- Peter LeFanu Lumsdaine <plumsdai@andrew.cmu.edu> wrote:

> >> [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.] 
> 
> Surely the intriguing thing here is not (as I understand you to be
> suggesting) the set-centricity that they're imposing, but rather that they're
> not imposing it as far as usual? 

Actually, what I find intriguing is that it is the definition of 
enriched category which seems to have priority over the definition
of internal category.  There are, I suppose, historical reasons for 
this (pre-1960 the focus tended to be on AbGp-enriched categories)
---but I think it fair to say that (for as long as I can remember, 
which obviously isn't that long from a "historical" perspective)
the majority of category theorists tend to adopt the internal 
category style of definition (of category) as more primitive.  

The issue at stake may seem minor: do we think of a class of arrows 
(which can later be partitioned into homsets), or do we think of the 
homsets first (and take their disjoint union later)?  But perhaps 
the fact that one group of people prefers one approach and everyone
else the other is symptomatic of a psychological divide? 

It's also worth noting, perhaps, how flukey it is that in the case 
V=Set, V-internal and small V-enriched categories happen to coincide.
Consider V=Cat, for example.  Or, note how different the requirements 
on V are, for V-internal and V-enriched categories to be defined. 

>When asked to define pretty much any
> algebraic gadget, most mathematicians will define a model of that algebraic
> gadget in Set (see e.g. en.wikipedia.org/wiki/Group_%28mathematics%29 ).

It is true that one would expect set-theoretic conservatives to deal with
small categories (~internal categories in the case V=Set), and more flexible
mathematicians to use arbitrary large categories (~internal categories, where 
V is a category of "large sets", or classes).  This only re-inforces the 
points made above.

Cheers,
Jeff.

Re: Teaching Category Theory

[Jeff Egger]

--- Bas Spitters <spitters@cs.ru.nl> wrote:

> > > You see what I really have in mind is not so much topos theory
> > > (which you might have suspected at first), but FOLDS.=20
>=20
> Could you give me a link to more information about FOLDS?

Sorry for not explaining!  FOLDS is an acronym for First Order Logic=20
with Dependent Sorts, with which I knew that Tom is familiar (having=20
discussed its pros and cons with him back when we were both students).

> Google was not very helpful.

Googling the whole phrase does produce satisfying results,=20
but to save you the effort, I can point you (all) towards
  http://www.math.mcgill.ca/makkai/folds/

Cheers,
Jeff.



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Re: Teaching Category Theory

[Steve Vickers]

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
>
>
>
>
>
>
>

Re: Teaching Category Theory

[Vaughan Pratt]

> 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

Teaching Category Theory

[Tom Leinster]

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