Subject: categories: Getting rid of cardinality as an issue (correction)

From: Vaughan Pratt <[email-masked]>
Date: Wed, 21 Apr 2004 23:15:13 -0700

[Note from moderator: apologies to Vaughan for missing his requested
change: 1 has been changed to 0 5 lines from bottom, so it reads:
`discrete 0-category'.]

Encouraged by the lack of objections to my previous message about why
Russell's Paradox should not be a big deal, I had a shot at shrinking the
position I spelled out there down to one paragraph, as follows.

------------
We shall axiomatize certain 1-categories using 2-categories.  We avoid
Russell's paradox by treating any aggregation of $n$-categories as an
$(n+1)$-category, and allowing for the possibility that the
$(n+1)$-category
$n$-$\CAT$ of all $n$-categories might be exponentially larger than any of
its members.  We impose no other size constraints besides the obvious
one of keeping things small enough to remain consistent.  Sets are defined
as usual as 0-categories and categories as 1-categories.
------------

While I'm happy to field objections like "too flippant", I'm more concerned as
to whether there are any technical flaws, and to a lesser extent philosophical
or religious concerns.  (I would not want to be held responsible for guns
being brought to the next UACT meeting if ever there is one.)

Makkai and Pare address the same issue in AMS CM 104 (Accessible Categories)
with a hierarchy of Grothendieck universes (three, since they like me stop at
2-categories for the application at hand).

Now the Grothendieck hierarchy is stepped through via ZF rather than Z, with
Fraenkel's Replacement axiom doing the heavy hitting.  This creates gaps
mind-bogglingly larger than my teensy exponential gaps above.  The general
idea seems to be that these gaps ought to be large enough to take care of
Russell while still not running headlong into inconsistency.  However gaps
this large do entail a certain amount of finger-crossing, and one might
question the logic of hitting Russell with a nuclear weapon that might send
some fallout your way when a harmless little tack-hammer will take him out.

One objection I can readily imagine to the above is that I've conflated
the n-category hierarchy with Russell's proposal for a ramified types
hierarchy.  I would disagree with that.  All I have done is to insist
on two things that seem to me to be independent.

1.  I have proposed to call aggregations of n-categories (n+1)-categories.
Now morphisms between n-categories are n-functors, and where there are
n-functors there are n-natural transformations, so this is hardly a bold
proposal.

2.  *Some* gap is needed between n-CAT and (n+1)-CAT, starting with the
requirement that Set be bigger than any set.  Russell's paradox is no
respecter of n, applying just as effectively to an (n+1)-category of
n-categories as it does to a 1-category of sets.

Certainly I have juxtaposed 1 and 2, but that is not the same thing as
conflating them.  Their mere juxtaposition provides sufficient armor
against both Russell's paradox and the Icarus risk of flying too close to
an inconsistently large cardinal.

The "prior art" for dealing with these issues has given rise to the adjectives
"small", "large," "superlarge", etc. and the nouns "set" and "class."
A good test for any revolution is the amount of blood it needs to shed.
The following definitions are aimed at minimal upheaval through maximum
compatibility with the status quo.

* An object is n-small when it belongs to an n-category.

* Small = 1-small, large = 2-small, superlarge = 3-small, etc.

* A set is a discrete 0-category.

* A class is a discrete n-category for unspecified n.

Hopefully Sol Feferman will give an even simpler solution in his talk
tomorrow.

Vaughan Pratt