FOM: small category theory

[Carsten BUTZ]

Dear Steve,

On Wed, 5 May 1999, Stephen G Simpson wrote:
> [...]  A category of size kappa where kappa is inaccessible cardinal
> is still set-size, i.e. small.  Isn't it?
> 
> Or is it?  Maybe we need to clarify the term ``small''.  Throughout
> this discussion, when I said ``small'', I always meant set-size.  Did
> you mean something else?
> 

There might indeed be some confusion (and category theory might be to
blame for it). As you guessed, small is not always set-size. Category
theorists pay attention to the distinction small/large, which in
sets/classes means indeed set/class, but if you choose to "implement" this
distinction with an inaccessible cardinal then small means strictly
smaller then that cardinal, and large means set.

The confusion arises only in discussions with set-theorists (and I am not
blaming those people). It is, admittedly, slang.

> Perhaps category theorists are accustomed to deliberately ambiguating
> on the term ``small'', sometimes meaning ``set-size'', other times
> meaning ``of size less than kappa where kappa is some fixed
> inaccessible cardinal''.
> 
> I have no objection to this kind of deliberate abuse of language, so
> long as it does not contribute to confusion regarding serious
> f.o.m. issues.  

It can, unfortunately.

> But such confusion seems to be a real possibility.  An
> example of such confusion is McLarty's phony claims concerning an
> alleged need for inaccessible cardinals in algebraic geometry and
> number theory.
> 

Why warming up this?
These issues were settled for this list weeks ago: Is there a need for
inaccessible cardinals for standard results in number theory? (Probably)
no. Do people use them (implicitly) for convenience? Yes they do (ask them
why they do this, not me). 

> 
> My point is as follows: 
> 
> Since category theorists are interested in ``smallness conditions'',
> and since ``smallness conditions'' are only special cases of
> cardinality conditions and the set/class distinction, all of which
> come directly from set theory, it would probably be a good idea for
> category theorists to pay more attention to the set-theoretic
> foundations of their subject.
>

They do, and I don't see you problem here. Just because no book on
category theory starts with the sentence: "This is our foundation: we work
in ZFC + there exist 2 inaccessible cardinals"
does not mean that category theorists are not aware of the fact that
something like this is needed to formalize their arguments. 
 
> In this context, I find Johnstone's deprecation of set-theoretic
> distinctions and results quite irritating.  And apparently there are
> other category theorists who share Johnstone's attitude of arrogant
> sloppiness vis a vis set theory.

There are two different attitudes: The first is indeed arrogance towards
foundations, the second is that the author of a paper/book does not fix a
particular foundation since it doesn't matter, as long as it has a
set/class (or small/large) distinction. I'm quite sure that Johnstone
belongs to this second group, and it is certainly not arrogance that
motivates (t)his choice.
(By the way, he wrote a (short) textbook: Notes on set theory and logic,
around 1987.) 

> 
> Yes, that's what I literally asked for, but in context what I really
> wanted was a deep theorem in category theory concerning ``smallness
> conditions''.
>

Well, not a deep one, but one where smallness conditions are essential, is
Freyd's adjoint functor theorem:
  Let A,B be categories which are locally small (i.e., their hom-classes
  are sets). Suppose moreover that A has all small limits.
  Let F: A -> B be a functor. 
  Then F has a left-adjoint if and only if 
  (i) F preserves small limits; and
  (ii) (The solution set condition)
  For every b in B there is a _set_ S(b) of arrows b--> F(a) such that
  every arrow b --> F(c) factors as b--> F(a) -------> F(c)
                                              F(i)
  for some i: a--> c in A and b--> F(a) in S(b).

The smallness condition is here essential, I quoted Freyd's comment on
this theorem about a week ago (you remember, the condition that S(b) is a
_set_ is not baroque. ...).

You where looking for something like this?

  Best regards,

  Carsten 

-----------------------------------
Carsten Butz
Dept. of Mathematics and Statistics
McGill University, Montreal, Canada