🚀 go-pugleaf

In a previous article, ziane@nuri.inria.fr (ziane mikal -) says:

              (much else deleted)
>
>Also, I don't know anything yet about categories. I have heard
>though that the category of all the categories is not a pb.
>Does category theory have similar paradoxes though ?
>
>
>Mikal Ziane.
>

	The category of categories is an open problem.  Set theoretic
approaches to it are not promising.  I have a tiny note in _JSL_ _57_
(pp.555-6) showing in effect that if you define categories set
theoretically in a set theory which includes a set of all sets, then
you can get an actual category of all categories but it will have
such terrible properties as to be useless.  (That does not mean that 
set theoretic considerations can not bear on the question, but 
straightforward set theory will not solve it.)

	The category of categories can be approached categorically too.
The classic source is by Lawvere, "The category of categories as
a foundation for mathematics" in _Proc. of the La Jolla conference on
Categorical Algebra", Springer-Verlag 1966 (pp.1-20).  More recently 
there is my own "Axiomatizing a category of categories" in _JSL__56_
(pp.1243-60).  Both briefly discuss what would be needed for "a
category of ALL categories".

	On a categorical approach a "category of ALL categories"
would presumably use Benabou's approach via fibrations (see Benabou 
"Fibered categories and the foundations of naive category theory" 
_JSL__50_ pp.10-37).  It would require a kind of universal fibration.  
It would require a universe of categories and functions, one of whose 
functors was a fibration such that:

	Every category in that universe is a fiber of that fibration

	Every functor in that universe appears as the functor
		between fibers induced by some arrow of the base
		of the fibration.

That second condition seems to me to want an elegant statement, and I
don't have one for it.

	It seems very unlikely that such a "category of all
categories" will give anything like a Russell's paradox--in part
because we never say EXACTLY WHAT any category is MADE OF.  Such
talk is nonsense on a categorical account.  So we can not define
a paradoxical category by giving an impossible requirement on the
categories and functors it should be MADE OF, the way we do for
elements of the Russell set.

	Such a fibration might lead to some kind of cardinality
paradox.  I am inclined to doubt it.  But I will speculate no more
here.  The problem remains open--even as to its exact statment.

Colin McLarty