FOM: Category of ALL categories
Colin McLarty
cxm7 at po.cwru.edu
Tue Feb 22 15:21:53 EST 2000
Steve Simpson has written:
>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. Or is it just word of mouth among
category theorists at Penn State?
Andrej.Bauer at cs.cmu.edu is right, for the most part, saying that
>When a category theorist speaks of ``the category of all
>categories'' (or more technically ``the 2-category of
>categories'') what is meant precisely is ``the large category
>of all small categories''. Rarely they mean something larger
>than that, say, the super-large category of large categories.
Of course there are technical variants, using Grothendieck universes for
example, but they all work to this effect.
However there is also some work on the idea of a category of all
categories, which would be one of its own objects. As Andrian-Richard-David
Mathias points out, set theorists have also looked at this:
>11. Some years ago I saw a preprint of a paper by Feferman
>investigating the possibility of a category of all categories
>within the framework of Quine's system of New Foundations,
>which admits a set of all sets. Can anyone tell me more about that ?
I would like to see Feferman's paper. But I can tell you how it
works out: There is a category of all categories in NF. The obvious
definitions involved are all stratified. And this category is unworkable,
because the awkward handling of functions in NF prevents the category being
cartesian closed. Details are in my article
Failure of cartesian closedness in NF, JSL 57 (1992) 555-56.
In fact, there is a little ambiguity about "functions" in NF,
because the various definitions of "ordered pair" which all work the same in
ZF do not all work the same in NF. But on any reasonable definition of
"ordered pair" and so of "function", for any two sets B and C there is a set
B^C of all functions from C to B; but there is generally no evaluation
function ev:B^CxC-->B taking each function f and element c to the value
ev(f,c)=f(c). So the category of sets and functions in NF is already not
cartesian closed.
If you define categories in terms of sets, then the collection
(whether or not it is a set within your set theory) of categories and
functors will inherit a lot of properties from sets and functions,
obviously. And in particular if the sets&functions are not cartesian closed,
the categories&functors will not be either.
Conversely suppose you require that sets and functions have the kind
of nice properties they have in ZF, say, as against NF. Specifically,
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".
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.
To use this framework is like using some variant set theory like NF, except
that this framework probably facilitates different kinds of restrictions
than you get just by restricting comprehension principles. It has not been
pursued much to my knowledge though there is new work on fibered categories
that I have not followed closely.
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