FOM: small category theory

[Carsten BUTZ]

Some minor comments on Steve's posting (not answering all his questions): 

> 
> 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.)

Maybe, maybe not. Some things will be lost (you probably loose some
adjoints that only exist because there is no limitation in size); as far
as applications are concerned you are probably right.

> 
> 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.

That's how category theorists think (often) about categories, small or
not. There is the fixed technical term of algebraic structures (like
groups, rings, distributive lattices, etc, but not local rings, fields,
etc.). (Small) categories do not belong into this group, but there are
what are called _essentially algebraic structures_, which roughly means
that there might be some operations (here: composition) which are not
total, but defined on a definable sub-type. This class of structures
behaves very much like algebraic ones (forgetful functors have adjoints,
i.e., "free" structures exist, etc). For (small) elementary toposes it was
Peter Freyd who first mentioned that they form an essentially algebraic
structure. 

> 
> 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.
> 

Since many (abstract) category theory theorems do not mention smallness
most of them have automatically a small version (like some of the examples
you mention below), this is (hopefully) wellunderstood among category
theorists. The same applies to the fact that often you get the "large"
version from the "small" ones, simply by exhausting the large category by
small ones. But, as said above, this can give problems.

It should be said as well that by restricting to small categories (like
countable modules over some countable ring, etc) you always have to
include some cardinality considerations, which often simply takes the
beauty of the results (but, fair enough, some people might say that there
is no beauty because there are no theorems because there is no
universe/inaccessible cardinal, ....)

[...]
> 
>    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.
>

Such things are more or less well known, after all, you don't need
smallness here but the same applies if you have a functor between large
abelian categories...

> 
> 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 ....
> 

Yes, and in fact, good guess. But, sorry, if I remember correctly (Colin
McLarty has a paper about this (?)) the category of all categories does
exist in NF, but does not have nice properties. NF doesn't help here.

  Best regards,

  Carsten 

-----------------
Carsten Butz
Dept. of Mathematics and Statistics
McGill University, Montreal, Canada