FOM: small category theory

[Till Mossakowski]

Stephen G Simpson wrote: (Fri, 30 Apr 1999 17:09:25)

>  > To enlighten this point, let me introcude a more explicit
>  > formulation of the above theorem:
>  >
>  > Let kappa be an inaccessible cardinal.
>  > 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.)
>  > - 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.
> 
> Doesn't this hold for a larger class of cardinals?  How about regular
> infinite cardinals, or limit cardinals, or strong limit cardinals?  If
> it holds for one of these classes, then this small category version of
> the theorem actually seems to be more general and more informative
> than the large category version.

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 can be seen very easily: Let lambda < kappa and card(Hom(A,B)) >=
2.
Then card(Hom(A,B^lambda)) >= 2^lambda, so 2^lambda < kappa.

Moveover, I guess that (I am not at my office and cannot check
it now) regularity of kappa is needed in the proof of theorem
(or at least in other important theorems).

>  > It is because in category theory, structural properties are intimately
>  > linked with smallness conditions (i.e. the distinction between
>  > "of size < kappa" and "of size <= kappa"), while this seems
>  > not to be the case for other branches of algebra.
> 
> But set-theoretic or cardinality considerations also come up in other
> branches of algebra, once you start looking at uncountable algebraic
> structures.  For instance, there are Shelah's famous set-theoretic
> independence results on the Whitehead conjecture for uncountable
> Abelian groups.  I don't think category theory has anything nearly as
> profound as that.

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

Till Mossakowski