FOM: Category of ALL categories

[Colin McLarty]

Steve Simpson wrote on 2 Mar 2000 (unless that is only when I got it on FOM
digest)

>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.

        The sets of ZF form a topos. The sets of NF do not. Some versions
of "recursive sets" do and some do not. Among set theories where sets do
not form a topos, some give a "category of all categories" in the most
naive sense, such as NF. This remark does not bear on those theories
(though my later paper does). This remark limits what we can get from any
set theory where sets form a topos.

	For example, some people have asked me if anti-foundational set theory AFA
would help give a "category of all categories", since it has sets that are
members of themselves. But this feature of membership is immaterial. The
sets of AFA form a topos and so the remark applies. 

> ``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.

        Yes. One way categories would not all have sets of objects, is if
we allowed proper classes of objects. But we also know that raising size
limits, by itself, will not bring us closer to a "category of all
categories". It brings us to the idea of Grothendieck universes.

	The simpler way for a category not to have a set of objects, is for it to
be a category in a first order theory of "categories and functors" with no
concept of "sets of objects" or "sets of arrows"--or perhaps a theory in
which the concepts are expressible but there is no proof that every
category has such sets. The first is the case for the earlier axioms in my
paper. As I recall, the latter is the case for the rest. The paper may
consider an axiom saying each category has set of arrows, but if so it
appears late in the development and you can consider the theory without it.
I don't have the paper with me now.

	
>``Every set of
>objects and arrows with a category structure corresponds to a
>category.''  This follows from the usual definition of ``category''.

        Yes, but even a set theoretic foundation for category theory could
modify it, e.g. by proposing that we only consider categories where the set
of arrows is no larger than the powerset of the set of objects or some more
arcane restriction. I am not interested in such tactics. But I have not
proved they will never work, and so my remark does not apply to them. 

	My paper looks at axioms which avoid this assumption in the simplest way:
a first order theory of categories and functors, in which we can prove that
if a category C has a set of arrows and a set of objects then these sets
have a "category structure" on them derived from C. But we cannot prove
every pair of sets with a category structure corresponds to a category.

 
>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?

	Benabou's paper talks about categories fibered over any category, not only
over sets or a topos. Historically, one of the motivations is the category
of modules, fibered over the category of commutative rings. The category of
commutative rings is not a topos. 

	The point is that the notion does not presume set theoretic definitions.
Benabou's methods have an obvious interpretation in any reasonable first
order theory of a "category of categories"--regardless of any intent to
comprise "all" categories. He does not talk of a "category of all
categories". But his methods seem to me the currently promising way to do so.

	How would a first order theory of categories and functors state that a
certain category C was the category of all categories? To me, the
interesting approach is to shift the question to fibrations: How would the
theory state that a certain fibration U-->C was a fibration of ALL categories?

	At the very least we require that for every category K there is a functor
K*:1-->C such that K is the fiber of U over K*. I.e. making this a pullback:

					K------>U
					|	|
					|	|
					v	v
					1------>C
					    K*

In other words K* is the object of C corresponding to the category K, and
each category has a corresponding object of C. And we require for each
functor f:K-->K' there is a functor f*:2-->C, in other words an arrow of C,
corresponding to f. I have never worked out exactly how much to require.

	Should we begin by requiring some property of all functors to C, which
will imply these properties of the ones from 1-->C and 2-->C?

	It is very little explored and I have no strong feeling that it will
produce an interesting "category of ALL categories". I am sure that
foundational aspects of fibered categories, and most especially Benabou's
ideas on equality and existence, are worth pursuing; and this could make a
good test case.    			

best, Colin