FOM: Category of ALL categories

Stephen G Simpson simpson at
Tue Feb 22 16:11:21 EST 2000

Colin McLarty Feb 22 2000 writes:

 > >When category theorists speak of ``the category of all 
 > >categories'' (as they often do), they are whistling in 
 > >the dark.
 > I would appreciate citations on this.

You are going over old ground here.  I gave some citations in the
discussion last May here on FOM.  One was to MacLane's book
``Categories For the Working Mathematician''.

 > Or is it just word of mouth among category theorists at Penn State?

So far as I know, there are no category theorists at Penn State.

 > suppose sets and functions form a topos. Then just as no set contains all
 > sets&functions, no set will contain all categories&functors. That remark is
 > on the first page of my article 
 > Axiomatizing a category of categories JSL 56 (1991) 1243-60
 >         In that context, both sets and categories are understood "up to
 > isomorphism", so the remark includes the theorem Steve Simpson has also
 > proved saying such a set theory will include no category of categories
 > containing *an isomorphic copy of* every category. I did not think of it as
 > a new discovery in 1991. However I would be happy to call it "Simpson's
 > theorem".

Thanks for that reference.  I will look up your article.  However,
according to what you are saying above, your result assumes a
set-theoretic framework satisfying the topos axioms, so isn't it
necessarily less general than my result, what I called my Russell
paradox for category theory (FOM, May 11, 1999)?  My result is in a
``naive category theory'' setting, with very few assumptions and no
set-theoretic framework.

However, even if my result is indeed new, I don't want it to be called
``Simpson's Theorem'', because there are many other theorems that I
would prefer to be known for.

 >         To me, the only current reason for interest in a "category of all
 > categories" is that a purely categorical approach might work, based on
 > Benabou's ideas of definability in
 > Fibered categories and the foundations of naive category theory JSL 50
 > (1985) 10-37.

Again, thanks for this reference.  Let's see if my version of
Russell's paradox for category theory works in Benabou's setting.

-- Steve

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