FOM: report from expert

[Colin McLarty]

Friedman cites an anonymous expert to say:

>I have been told that there is absolutely no trace back from the references
>used in the body of the Wiles paper to Universes (of Grothendieck).

        The first reference in Wiles's bibliography is to Altman and Kleiman
INTRODUCTION TO GROTHENDIECK DUALITY THEORY (Berlin, New York,
Springer-Verlag, 1970). The book freely quantifies over "Abelian categories"
and "functors" and even defines certain functors by properties that
themselves quantify over functors. These categories and functors are, from
the usual ZF point of view, proper classes. If they exist as sets, then
universes exist. Naturally Altman and Kleiman are unconcerned with
foundational issues.

        These are the basic methods of derived functor cohomology. Wiles,
and everyone else who use cohomology in number theory, use these methods. I
don't know if Wiles cites Altman and Kleiman in the body of his work or the
introduction, but I do know his work relies on many commonly used methods he
describes in the introduction.  

Friedman says his expert told him:
        
>that in any serious use of category theory in number theory, one must
>control the categories involved in order to accomplish anything, and a tiny
>part of that control is knowing that the cardinality of the categories is
>small.

This is quite correct, one must control it. But not ALL the categories need
be small, only the crucial ones. E.g. in the Tate example quoted earlier,
the category of commutative Hopf algebras over a given ring has no small
cardinality, it is a proper class in the usual ZF approach--but in practice
you will not use the whole category of these Hopf algebras. The basic
methods are usually developed using large categories, and then specific
applications come about by finding suitable small ones to relate with these. 


>In particular, I have been told that it is grossly misleading to consider
>any connection whatsoever between Universes and serious number theory of
>the integers, rationals, or finite degree extensions of the rationals.

It is absolutely true that for a top career in number theory you should not
worry about Grothendieck Universes, or any other foundational questions. You
will, of course use methods that have foundations. And if you go so far out
on a limb as to cite any standard reference on homological algebra with
derived functors, then you refer to Grothendieck universes.

>In my opinion, the Editorial
>Board and/or the referee would require that the use of Universes be
>explicitly mentioned.

        This is false of book editors, as shown by the examples I cited of
Tamme and Cornell, Silverman et al.