FOM: small category theory

[Stephen G Simpson]

Till Mossakowski 01 May 1999 13:25:18 writes:
 > >  > Let C be a category of size <= kappa that
 > Oops. I forgot to add:
 >      - has hom-sets of size < kappa
 > (this is left implicit in the Herrlich-Strecker definition of
 >  category, so I overlooked this point. Sorry.)

Is the theorem still true without the assumption ``hom sets of size
less than kappa''?

Your comment on Herrlich-Strecker is not very clear.  I would be very
surprised if Herrlich-Strecker include a phrase such as ``hom sets of
size less than kappa'' in their definition of ``category''.  Perhaps
what you mean is that they define a category to have small hom sets,
i.e. it is required that for all objects A and B, Hom(A,B) is a set.
But the reason for this restriction is not very clear, given that
Herrlich-Strecker probably insist on allowing the categories
themselves to be proper classes.  And in any case, being a set is not
at all the same as being a set of size less than kappa.

It seems to me that it would be a good idea for category theorists to
take account of standard set-theoretic notions (cardinality etc) in
order to clear up all this confusion and obtain sharper theorems.  Why
are category theorists apparently reluctant to do this?

 > >  > - 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.

 > If kappa is a cardinal and C a category satisfying two of the
 > premises of the above theorem (namely C has hom-sets of size < kappa,
 > and C has limits of size < kappa), and moreover, C is non-trivial
 > in the sense that two objects with more than one morphism between
 > them exist, then kappa is a strong limit cardinal.

This remark loses its force if the assumption ``hom sets of size less
than kappa'' is superfluous.  Once again, is the theorem true without
that assumption?

 > I do not know Shelah's results, but there are deep results in
 > category theory as well.

Could you please state a deep result in category theory?

 > Category theory differs from other branches of algebra in the point
 > that you very quickly have to consider smallness conditions (a book
 > on category theory quoting only those results not involving
 > smallness considerations would be very poor).

This may be the case.  However, it doesn't follow from this that
category theory has obtained any deep insight into ``smallness
conditions''.  It seems to me that set theory has obtained and can
continue to obtain much deeper insights, expressed in terms of
cardinality conditions.  After all, ``smallness conditions'' are only
a special case of cardinality conditions.  Notions such as cardinality
and the distinction between sets and proper classes were first
formulated in the context of set theory, not category theory, and it
seems to me that they are best studied in the set-theoretic context.

-- Steve