FOM: Category of ALL categories

[Stephen G Simpson]

This is a followup to postings of 22 Feb 2000 on the same subject, by
McLarty and me.

I think McLarty is still holding out hope for some sort of categorical
foundation of category theory in terms of an alleged ``category of all
categories''.

As promised, I have now looked up the articles by McLarty (JSL 56,
1243-1260) and Benabou (JSL 50, 10-37).

The relevant remark on the first page of McLarty's article is:

  ``There cannot be an actual category of all categories if sets form
  a topos, every category has a set of objects, and every set of
  objects and arrows with a category structure corresponds to a
  category.''

I am trying to understand this remark, but the assumptions seem
somehow odd.  ``Sets form a topos.''  Well, sets are the standard
example of a topos, so this is well known.  ``Every category has a set
of objects.''  This is wrong if we consider categories with a proper
class of objects, e.g., the category of all groups.  But it is right
if we restrict attention to set-size categories.  ``Every set of
objects and arrows with a category structure corresponds to a
category.''  This follows from the usual definition of ``category''.

Perhaps McLarty could explain his remark further.  I am interested in
understanding it, in order to compare it to my ``Russell paradox for
naive category theory'', FOM, 11 May 1999.

Benabou's paper is an interesting informal discussion of fibered
categories over the category of sets, and possible generalizations
where the category of sets is replaced by a topos.  But I have not
found where the paper alludes to the possibility of an alleged
``category of all categories''.  Am I overlooking something?

Incidentally, I think I can answer Benabou's question 7.3.  Let M be a
model of ZF, and let S be a subset of M.  In Benabou's terminology, S
is *representable* if there exists b in M such that S = {a in M : M
satisfies ``a in b''}, and S is *definable* if for every representable
X, S intersect X is representable.  Benabou asks whether there exist
models M of ZF such that every subset of M is definable.  It seems to
me that this is the case if and only if M is isomorphic to a
transitive model of ZF consisting of all sets of rank less than alpha,
for some limit ordinal alpha.  Note that alpha need not be an
inaccessible cardinal.

-- Steve