FOM: Foundations of category theory

[Till Mossakowski]

Stephen G Simpson wrote Wed, 28 Apr 1999 20:25:08 :

> As McLarty has noted, Corry speculates on page 332 that Bourbaki
> stayed away from categories because it's difficult to reconcile them
> with structures.  In actual fact, such a reconciliation is difficult
> only if you arbitrarily decide that, while structures must be sets,
> categories may be proper classes.  There is no serious mathematical
> reason forcing anyone to make such a decision, so Corry's speculation
> seems incorrect.

Category theory is largely concerned with the study of smallness
conditions, so the distinction between classes and sets (and
categories possibly being classes) is crucial for category theory.
I wonder whether category theory can be founded entirely on
sets. It would be nice, because the real interest of category theory
is the study of structures, and the smallness conditions only
come in since one is forced to study them (which, by the way, 
will not change when ZFC is used as foundation - but at least
the assumption of the existence of a universe would go away).
Surely, not all of category theory can be developed within such a
setting, 
but perhaps a large relevant portion?

The more exciting story would of course be a foundation that
would allow to reduce the amount of study of smallness conditions
within category theory, and that would allow to concentrate on the
structure.
But this seems to be heavily difficult, if not impossible.

Till Mossakowski