FOM: small category theory

[Stephen G Simpson]

Todd Wilson 5 May 1999 16:45:44

 > Without category theory, we would not be able to express in a
 > sufficiently final way, for example, that limits for all diagrams
 > can be constructed from products and equalizers (or many other
 > similar results).

Perhaps.  However, *small* category theory (i.e. the theory of
set-size categories) would equally well suffice to express these
results, and in an equally ``final'' way.  By sticking to *small*
category theory, category theorists would be able to live perfectly
well within the current standard set-theoretic f.o.m. setup,
formalized as ZFC.  There is no need for them to drag in a lot of
superfluous notions such as small and large, Grothendieck universes,
etc.  There is no sound mathematical reason for them to do so.

 > when a category theorist sees a structural result like the Yoneda
 > Lemma ..., he or she wants to highlight those simple structural
 > properties without having to clutter the picture with not-so-simple
 > notions from set theory -- especially notions involving certain
 > classes of (large) cardinals.

What in the world are you talking about?  As I pointed out earlier,
Yoneda's lemma can be stated perfectly adequately in the standard
set-theoretic f.o.m. setup, in terms of small (i.e. set size)
categories, with no loss of content or elegance.  And the same goes
for most or all other theorems of category theory.  But it seems the
category theorists will have none of that.  Instead, it is they who
insist on cluttering up the picture, claiming to need additional
superstructure including the collection of all functions from a proper
class to a proper class, arbitrarily large inaccessible cardinals,
etc.  (See for instance McLarty 23 Apr 1999 08:55:14 and Mossokowski 5
May 1999 00:04:05.)

 > If current set-theoretic foundations do not allow these simple
 > structural results to be expressed in their most elegant form, the
 > one is led to look for alternative foundations for which this is
 > possible.

OK, let's assume you are right.  Let's assume the category theorists
really do think they have some serious mathematical reason for being
unhappy with the current set-theoretic foundational setup.  And they
want to look for an alternative foundational setup.  OK, fine.  But
then, it's reasonable to ask: What have they accomplished along these
lines?  It seems to me they have only borrowed some concepts from set
theory and restated them in their own muddled language.  [ I refer to
the small/large distinction (a muddled, watered-down version of
cardinality and the set/class distinction) and Grothendieck universes
(a muddled version of inaccessible cardinals). ]  And in the process
they have managed to spread confusion about some serious
f.o.m. issues.

To strike a positive note, let me say again that I have a lot of
respect for category theory as an organizationaling tool in various
branches of mathematics: algebraic topology (Eilenberg-Steenrod),
homological algebra (Eilenberg-MacLane), algebraic geometry
(Grothendieck), and maybe some areas of theoretical computer science.
My criticism is directed only at the *foundational* pretensions of
category theory.

-- Steve