FOM: H-S appendix; small category theory

[Stephen G Simpson]

The foundational setup in the Herrlich-Strecker book is the same as
that of the Adamek-Herrlich-Strecker book
(sets/classes/conglomerates).  The goal is modest: ``a foundation that
is sufficiently flexible so as not to unduly inhibit our categorical
inquiry and one that we can be reasonably sure will not lead to
paradoxes.''

In addition, H-S has a 4-page appendix on foundations.  Four
foundational approaches are compared.

1. VNBG.  ``small'' = set, ``large'' = proper class.  The authors say
   that this setup has disadvantages in that functor categories [A,B]
   cannot be formed if A is large, and that the Yoneda lemma cannot be
   stated properly.  

   (But it seems to me the Yoneda lemma *can* be stated properly.  See
   my posting of 5 May 1999 17:24:51.  Also, the VNBG approach has
   been used many times, e.g. in Eilenberg-MacLane.  Also, the VNBG
   seems to be adequate for everything in Hartshorne's book; see my
   postings of 22 Apr 1999 17:42:45 and 23 Apr 1999 13:41:15.)

2. Grothendieck universes.  ZF plus a proper class of inaccessible
   cardinals.

3. Throw out set-theoretic foundations and replace it by categorical
   foundations.  Take ``category'' and ``functor'' as primitive
   notions.  The authors say that this approach is due to Lawvere.
   They mention the following disadvantages: (1) It has less intuitive
   appeal.  (2) It is not yet fully developed.  (3) To use it in an
   introductory text would lead to undue confusion.

4. The one-universe approach.  ZF plus one inaccessible cardinal.
   This is the approach that the authors opt for.  See my posting of
   19 May 1999 17:25:14.

But, for some reason, H-S never considers the straightforward approach
that I have advocated, i.e. the ``small category'' approach, where we
work in ZC, categories are sets, and there is no category of *all*
sets, *all* groups, *all* topological spaces, etc.  (See my posting of
29 Apr 1999 18:39:59.)

In other words, they never even consider the idea that categories may
have the same status as other mathematical objects, i.e. they are not
bigger or more inclusive.

Why do they refuse to consider this simple idea?  Are they holding out
for some hope of ``categorical foundations''?

-- Steve