FOM: small category theory

[Stephen G Simpson]

Till Mossakowski 30 Apr 1999 12:32:43 writes:
 > To enlighten this point, let me introcude a more explicit
 > formulation of the above theorem:
 > 
 > Let kappa be an inaccessible cardinal.
 > Let C be a category of size <= kappa that 
 > - has all colimits of size < kappa,
 > - has a separator, and 
 > - has quotient-object-classes of size < kappa,
 > Then C also has all limits of size < kappa.

Doesn't this hold for a larger class of cardinals?  How about regular
infinite cardinals, or limit cardinals, or strong limit cardinals?  If
it holds for one of these classes, then this small category version of
the theorem actually seems to be more general and more informative
than the large category version.

 > It is because in category theory, structural properties are intimately
 > linked with smallness conditions (i.e. the distinction between
 > "of size < kappa" and "of size <= kappa"), while this seems 
 > not to be the case for other branches of algebra.

But set-theoretic or cardinality considerations also come up in other
branches of algebra, once you start looking at uncountable algebraic
structures.  For instance, there are Shelah's famous set-theoretic
independence results on the Whitehead conjecture for uncountable
Abelian groups.  I don't think category theory has anything nearly as
profound as that.

 > As Carsten Butz mentiond, Colin McLarty has shown that NF is of not
 > so much use here, since the category of categories would not be
 > cartesian closed.

OK, so much for that idea.

Carsten Butz 30 Apr 1999 11:27:29 writes:
 > (1) Category theory _does_ want to speak about large categories, people do
 > want to study the category of all (small) groups, of all (small) vector
 > spaces over some fixed field, of all (small) topological spaces, etc.

Why?  Is there any compelling mathematical reason for category
theorists to insist on large categories?  I'm not convinced there is.
Maybe the only reasons are historical.  Or maybe the reasons have to
do with the desire of some category theorists to push the idea of
``categorical foundations'' -- an idea that I think is dubious.  It
seems to me that most or all serious applications of category theory
(e.g. in Hartshorne's book) could get by very well with small
categories.

 >   But, look at the "contra-variant" powerset functor
 >    P: Set^op --> Set
 > from the opposite of the category of sets, to sets. It sends a set X to
 > P(X), the set of all its subsets, and sends a map to its inverse image.
 > This functor has a left-adjoint, which is simply the same functor, now
 > viewed as a functor Set --> Set^op.
 > If we restrict P to some subcategory like Set(k), sets of cardinality at
 > most k, the above adjointness is clearly lost.

What if you look at the category of sets of cardinality less than k,
where k is a strong limit cardinal?  Is adjointness lost in this case?

Here again, small categories defined by simple cardinality
restrictions seem to be enough to get a good theorem.  We don't need
to go to large categories.

 > As for sets you can form (gramatically) the "category of all
 > categories" (this is different to other structures: the collection
 > of all groups does not have a group structure, ...).

But the collection of all semigroups has a semigroup structure (under
direct product, for instance).  So again, category theory is not
special or unique here.

-- Steve