# FOM: small category theory

Stephen G Simpson simpson at math.psu.edu
Wed May 5 11:52:42 EDT 1999

```Till Mossakowski 05 May 1999 00:04:05

> Let kappa be a regular cardinal.
> Let C be a category of size <= kappa that
>       - has hom-sets of size < kappa
>       - has all colimits of size < kappa,
>       - has a separator, and
>       - has quotient-object-classes of size < kappa,
> Then C also has all limits of size < kappa.
...
> To be able to apply the theorem in a useful context, we need that
> kappa is inaccessible, which goes beyond ZFC. ...

I'm not completely convinced of this, because (1) I don't know what
useful context this theorem can be applied in, (2) I think it may be
possible to generalize the theorem, replacing some of the occurrences
of kappa by some fixed cardinal lambda less than or equal to kappa,
and in this way the theorem may make sense even when kappa is not
inaccessible.

But OK, for the sake of discussion, let's take it for granted that
this is a useful theorem, and that the only reasonable way to state it
is in terms of inaccessibles.

> Thus, "small" category theory does not make much sense.

Why do you conclude that small category theory does not make much
sense?  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?

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.  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.

> Category theory has set-theoretic foundations, and it studies the
> interplay of smallness conditions with the structural properties
> (and not the smallness conditions for themselves).

Well said.

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.

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.

Till Mossakowski 5 May 1999 11:43:05 writes:
> Steve Simpson (Sun May  2 22:08:40 1999) asked for a deep theorem
> in category theory.

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''.

> I want to mention the following correspondence theorems

These are interesting and deep theorems of category theory.  Do they
involve ``smallness conditions'' in any serious or essential way?

-- Steve

```