FOM: small category theory

[Till Mossakowski]

Stephen G. Simpson wrote (Wed May  5 19:05:26 1999) :

> 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''.

Indeed, they work with (at least) one universe, and call
elements of the universe "sets", and subsets of the universe
"classes" (while arbitrary sets are sometimes called 
"conglomerates").
This terminology is justified by the fact that the "sets"
and "classes" within a model of ZFC + universe form a model
of VNBG.


>  > 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?

Yes, smallness conditions come in through the restriction of
the arity of the operations of the algebras. To restrict the arity
to kappa is equivalent to th requirement that the underlying functor
of the algebraic concrete category maps kappa-directed colimits
into epi-sinks. Now if there is no restriction on the arity
of operations, then a characterization is still possible, but
then concrete co-wellpoweredness (which is similar to co-wellpoweredness)
is needed as an additional assumption.
Note that compact Hausdorff spaces or complete lattices can
be only described as algebras with unbounded arity of
operations. Now compact Hausdorff spaces and complete lattices differ
essentially: free compact Hausdorff spaces exist, while
free complete lattices do not exist. This is directly connected
with the fact that compact Hausdorff spaces are concretely
cowellpowered, while complete lattices are not.
To be able to state and explain this phenomenon, you definitely
need the small/large distinction.

Till Mossakowski