FOM: Category of ALL categories
Stephen G Simpson
simpson at math.psu.edu
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
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
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.
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