FOM: Russell paradox for naive category theory

Carsten BUTZ butz at
Mon May 3 20:21:43 EDT 1999

On Mon, 3 May 1999, Stephen G Simpson wrote:

> As I think about this, it seems to me there is an analog of the
> Russell paradox for naive category theory.  I outline it below.

My words. Of course there is. I guess most people know this. And those who
don't (maybe some younger ones) have a surprisingly good feeling for
what is allowed and what not. I have never seen a paper in category
theory that actually runs into Russell's paradox or a variant thereof.
The "typing" in papers on category theory is usually ok explicitly or it 
can at least be restored. 

I just checked: Mac Lane's Categories for the
working mathematician for example does explicitly include a (short) 
section on foundations, pointing out the importance of the distinction
small/large or of variants of this distinction. Another example of this
awareness I take from Freyd, Abelian categories, p. 85, the emphasis is
his (it's about the adjoint functor theorem and the solution set
  The stipulation in condition two, that S_B is a {\em set}, is not
  baroque. Because mathematics has progressed for a long time without
  having had to take the set-class distinction seriously does not mean
  that the distinction is spurious. The requirement that there be a set
  such as S_B is of the same nature as the requirement that a group be
  generated by a finite set. Both requirements can be very difficult to
  fulfill, and both can have powerful consequences.

By the way, some people use Cat for the category of all small categories
(itself large). CAT for the category of all large categories
(iteself "superlarge" ), ... Some unfortunately leave out the specifying
adjectives "small", "large", etc. 

  Best regards,

  Carsten Butz

Carsten Butz
Dept. of Mathematics and Statistics
McGill University, Montreal, Canada

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