[FOM] FLT and ZFC, again

[Messing]

Concerning the remarks made by the number theorist who prefers to remain
anonymous, it seems necessary to fill in some backround and clarifying
remarks.  I would great appreciate it if this can be forwarded to the
anonymous number theorist.

Firstly, Grothendieck was not a logician or a set theorist.
Nevertheless as someone who from the mid 1950's onward was very involved
with Category Theory and its applications, he was sufficient aware of
set theoretic issues and took great care with regard to them.  In his
course on Category Theory in Algiers in November, 1965, he starts
Chapter 0, entitled Cadre Logique, as follows: "Lorsque l'on definit une
categorie, il y a des inconvenients a supposer que les objets forment
une classe, au sens de la theorie des ensembles de Godel-Bernays.  En
effet, si l'on sait definir les applications d'une classe dans une
autre, ces applications ne forment cepedant pas elles-memes une classe.
En particulier on ne saurait parler de la categorie des foncteurs
d'une categorie dans une autre."

For Grothendieck it was absolutely essential to consider, for two
categories C and D, the category Hom(C, D) whose objects are the
functors u:C --> D and whose morphisms are the natural transformations
f:u ==> v between two such functors.  This is of course not only
extremely natural, but is absolutely essential for discussing sites,
topoi, cohomology of sheaves, ....

As a working Arithmetic Geometer who has made extensive use of the
etale, crystalline and syntomic sites and their cohomology (as well of
course of other Grothendieck topologies such as the fppf and the fpqc),
it is clear that the language of universes introduced by Grothendieck,
is extremely convenient and natural.

I would like to take issue with a remark made by the anonymous
number-theorist: "but in practice **nothing would change** regarding
etale and fppf cohomology."  I do not know how much the anonymous
number-theorist knows about the fppf topology.  He is correct as far as
the etale topology goes because the topos associated to the small etale
site of a scheme is functorial in the scheme.  In other words, if for
any scheme Z, Et(Z) is the category whose objects are etale morphisms
g:W --> Z and whose morphisms are all Z-morphisms between two such
objects and if Z~ denotes the corrsponding topos, then if f:X --> Y is a
morphism of schemes, there is induced a morphism F:X~ --> Y~.  If
instead we took the category FPPF(Z) to have as objects all fppf
morphisms g:W --> Z and whose morphisms are all Z-morphisms between two
such objects and we again denote by Z~ the associated topos, then if
f:X --> Y is a morphism of schemes there is no morphism F:X~ --> Y~. The
reason is clear.  A morphism of topoi is, by definition a pair of
functors f_*:X~ --> Y~, f^*:Y~ -->X~, such that f^* is left adjoint to
f_* AND such that f^* commutes with finite inverse limits. There is
indeed such an adjoint pair, but f^* does not commute with finite
inverse limits in general.  The reason for this is that in the
categories FPPF(Z) fiber products of objects do not, in general, exist.
This is one reason why Grothendieck introduced the BIG sites in addition
to the small sites.  The big site of a scheme Z (for any topology)
consists has the underlying category having as its objects all morphisms
of schemes g:W --> Z. If Z~ denotes the corresponding topos then any
morphism f:X --> Y of schemes induces a morphims of topoi
F:X~ --> Y~.  For a fixed scheme Z, from the point of view of
cohomology, the big and small sites will give canonically isomorphic
results. This is the "chocolate exercise" of SGA 4.  But one is
interested in and needs more than just cohomological calculations for a
fixed scheme. Anyone who thinks in terms of the "six operation"
formalism of Grothendieck realizes this instaneously.

The anonymous number theorist seems to think that Grothendieck
introduced universes in SGA 4 because he was thinking of some possible
"exotic" Grothendieck topologies that might arise in the future.  This
is simply false.  Universes are introduced on page 1 of Expose I of
SGA 4 and are used throughout the 1583 pages of SGA 4.

The anonymous number theorist seems to think that it is because
(isomorphism classes of) coverings of a fixed object X of a site, might
not form a set is the reason Grothendieck introduced universes,  This is
simply false.  Long before one meets Cech cohomology in SGA 4, one needs
universes in order the have the associated sheaf functor defined.  Cech
cohomology is treated in Expose V of SGA 4, but is not a primary tool
for algebraic geometers in connection with the etale, fppf, ...
topologies.  Of course it is used, but rarely to calculate H^17 for some
abelian sheaf on some site.  Of course, in addition, Cech cohomology
will not in general coincide with the cohomology defined as the right
derived functors of the global section functor.  For this one needs, not
just covers, but hypercovers as well.  In the case of the etale
topology, it is a  theorem of Artin that for X a scheme quasi projective
over a ring, and F an abelian sheaf for the etale topology on X,
H^*(X_et,F) is isomorphic to the Cech cohomology of X_et with
coefficients in F.  This is the analogue of the fact that for X a
paracompact space, its sheaf
An attentive reading of the appendix to Expose I of SGA 4 shows that
Grothendieck and Bourbaki were well aware that the axiom, denoted by
(UA) in expose I of SGA 4, stating that every set is an element of a
universe is independent of the axioms of ZFC and its relative
consistency can not be proven.

Finally there is another axiom of universes introduced by Grothendieck,
the axiom denoted by (UB) of expose I of SGA 4, which is much less
discussed than the axiom of the last paragraph.  This depends upon the
logic employed in Bourbaki, which utilizes the notation tau_x for the
Hilbert epsilon symbol epsilon_x.  If P is a property of sets and U is
an universe universe U,  and there is an element y of U which satisfies
P, then tau_x(P) (the distiguished set which satisfies P) is an element
of U.  The use of this axiom is the following: an U-category C, is, by
definition, one such that for any pair of objects x and y of C, Hom_C(x,
  y) is "U-small" that is admits a bijection to an element of U.  If
U-sets denotes the category of sets which are elements of U and C^
denotes the category of contravariant functors F:C --> U-sets, that is
the category of U-presheaves on C, then, provided C is an U-category,
there is a natural functor h:C --> C^.  This is the functor that occurs
in Yoneda's lemma: for x and object of C and F an object of C^ there is
a natural bijection Hom_{C^}(h(x), F) --> F(x).  The construction of h
depends in an essential manner on the axiom (UB).

It is my impression that there is very little FOM discussion of either
Hilbert's epsilon symbol or of Bourbaki formulation of set theory.  In
particular the chapitre IV Structures of Bourbaki.  For reasons,
altogether mysterious to me, the second edition (1970) of this book
supressed the appendix of the first edition (1958).  This appendix gave
what is, as far as I know, the only rigorous mathematical discussion of
the definition of the word "canonical".  Given the fact that Chevalley
was, early in his career, a close friend of Herbrand and also very
interested in logic, I have guessed that it was Chevalley who was the
author of this appendix.  But I have never asked any of the current or
past members of Bourbaki whom I know whether this is correct.

William Messing
Professor, School of Mathematics
University of Minnesota