FOM: small category theory

[Stephen G Simpson]

Till Mossakowski writes:
 > Category theory is largely concerned with the study of smallness
 > conditions, so the distinction between classes and sets (and
 > categories possibly being classes) is crucial for category theory.

I'd like to follow up on these points.  What is a ``smallness
condition''?  Can you give some examples?  What major insights
concerning ``smallness conditions'' has category theory obtained?

So far as I know, the terms ``small'' and ``large'' in category theory
refer only to a distinction that was borrowed from set theory, namely
the distinction between sets and proper classes.  And maybe category
theory has made some use of some other distinctions that were also
borrowed from set theory.  For example, there is the distinction
between cardinality less than kappa and cardinality equal to kappa,
where kappa is some fixed inaccessible cardinal.  This is what
category theorists call Grothendieck universes.  This idea also came
from set theory, didn't it?  For example, MacLane's definition of a
Grothendieck universe in his book ``Categories for the Working
Mathematician'' is given in terms of set theory.

Is there a good way to get at these distinctions in purely
category-theoretic terms, not using concepts borrowed from set theory?
Did category theory obtain any insights not already obtained by set
theory?  Please enlighten me.

 > I wonder whether category theory can be founded entirely on
 > sets.

I'm not sure what you have in mind here, but let me mention a specific
proposal.  What would be lost, and what would be gained, if category
theory restricted itself to ``small'' (i.e. set size) categories?
Let's call this subject ``small category theory''.

It seems to me that, if we were to replace category theory by small
category theory, then essentially nothing would be lost, at least with
respect to applications.  (See also my posting of 22 Apr 1999
17:42:45, concerning how this would work in Hartshorne's book on
algebraic geometry.)

On the other hand, I can thing of at least two advantages that might
be gained:

1. A small category is just another kind of algebraic structure, like
   a group or a ring or a vector space.  So it would be possible to
   include an exposition of small category theory in with an
   exposition of the rest of algebra, in a very simple way.  Category
   theory would no longer stick out like a sore thumb.

2. In those cases when you really want to deal with ``large''
   (i.e. proper class size) categories, it seems to me you could state
   your theorems more informatively in terms of small categories, in
   such a way that they would immediately imply the ``large''
   versions, with additional information.

   For example, take the theorem asserting the existence and universal
   property of derived functors on Abelian categories with enough
   injectives.  (Hartshorne, beginning of Chapter III.)  You could of
   course specialize this to small Abelian categories with enough
   injectives, and nothing would be lost as far as applications.  But
   you could also jazz up the statement by making it more functorial,
   so it would then give information about large categories too.

   Here's what I am getting at.  The original statement would be
   something like: given a functor F:A->B where A and B are small
   Abelian categories with enough injectives, a derived functor F'
   exists and has a universal property.  The jazzed-up statement would
   say: given a commutative diagram

        F:A->B
          |  |
          v  v
        G:C->D

   there is a similar diagram for the derived functors F' and G', etc.
   (Maybe we need to assume that A->C and B->D are embeddings, and
   that injectives in A are injectives in C, etc etc.)

   And this jazzed-up theorem for small categories would immediately
   imply the original theorem for large categories, because a large
   Abelian category with enough injectives is easily seen to be a
   union (direct limit) of small Abelian full subcatgories with enough
   injectives.

   Maybe all this is well known, or trivial.  Or maybe it's completely
   wrong.  Please enlighten me.

Mossakowski:

 > The more exciting story would of course be a foundation that would
 > allow to reduce the amount of study of smallness conditions within
 > category theory, and that would allow to concentrate on the structure.
 > But this seems to be heavily difficult, if not impossible.

Do you mean you want something like the category of all categories?

Has anyone ever tried to set up something like this on the basis of NF
set theory?  I don't know enough about NF to know whether this makes
sense, but I seem to have heard that NF has the set of all sets ....

-- Steve