FOM: small category theory

[Stephen G Simpson]

Till Mossakowski 11 May 1999 23:10:12 

 > Indeed, they work with (at least) one universe, and call elements
 > of the universe "sets", and subsets of the universe "classes"
 > (while arbitrary sets are sometimes called "conglomerates").

This terminology is obviously inconsistent with standard set-theoretic
terminology.  Thus it invites confusion.  Why do category theorists
insist on doing this kind of weird stuff?

Some category theorists claim that category theory provides a unifying
framework for mathematics.  My comment: If they really want to help
unify mathematics, they could start by adhering to standard
set-theoretic terminology which was developed and standardized long
before categories were invented.

 > This terminology is justified by the fact that the "sets"
 > and "classes" within a model of ZFC + universe form a model
 > of VNBG.

No, this terminology isn't justified, because sets (i.e. what you call
``arbitrary sets'') and what you call ``sets'' (i.e. elements of
V_kappa where kappa is some fixed inaccessible cardinal) are two
different things.  To use this kind of inconsistent terminology, even
with quote marks, is to invite confusion.

Till Mossakowski 11 May 1999 23:18:47

 > Please have a look at Adamek, Herrlich, Strecker: Abstract and
 > concrete categories, p. 385. On this page, both small/large
 > distinctions and cardinality considerations (involving regular
 > infinite cardinals) appear.

But wait a minute.  Your remark above implies the small/large
distinction *is* a cardinality consideration.  Namely, according to
you, ``small'' means of size less than some fixed inaccessible
cardinal, right?  So it seems that this is consistent with my idea to
restrict attention to set-size categories.  Inaccessible cardinals may
then appear as a hypothesis, if desired.

Incidentally, I still think there are some authors who use ``small''
and ``large'' to refer to sets and proper classes in the sense of VNBG
set/class theory.  This is apparently inconsistent with
Adamek-Herrlich-Strecker, but it has the merit of being consistent
with standard set-theoretic terminology.

 > Appearantly, the authors are very aware of the set-theoretic
 > foundations. 

That's good!  And it is also consistent with other good things I have
heard about the Adamek-Herrlich-Strecker book.  I will have a look at
that book.

I never said that *all* category theorists have a funny or hostile
attitude toward set-theoretic foundations.  Only some of them do.  One
who does is Lawvere.  Another is McLarty, as evidenced by the
``categorical foundations'' debate of last year.  Another is
Johnstone, as evidenced by the preface of his topos theory book.

-- Steve