Subject: categories: models of SDG

From: [email-masked] (Eduardo Dubuc)
Date: Thu, 14 Sep 2000 18:16:43 -0300 (ARG)

GERM DETERMINED IDEALS AND WELL ADAPTED MODELS OF SDG
          
            I write this note because I feel it is time to let know 
explicitly to the category theory community the following situation, and 
to make a request.
            
            In a recent paper on SDG "Super smooth topoi, by H. 
Nishimura" one can read the following: "Therefore we must build our model 
theory of Synthetic Supergeometry not directly after the standard manner 
of Moerdijk & Reyes [1991] [*] but after the manner of Dubuc-Taubin 
[1983]". Now, this "standard manner" of building models was  developed by 
Dubuc  and not by Moerdijk-Reyes. 

(in [1983] I developed also the analytic model, which does not correspond
to an algebraic theory in the Lawvere sense, and can not be done as the
differential models which correspond to C-infinity rings). 

           [*][M-R] "Models of Smooth infinitesimal Analysis", Moerdijk 
and Reyes, Springer Verlag 1991.

            Of course, I understand that when a monograph is available, 
the proper reference is that, and not the original papers. But, phrases 
as the one quoted above are ambiguous, and at fault is not Nishimura.
            
            I shall resume the history of the subject: In a series of 
papers and many lectures given specially in Montreal, Sydney, Oberwolfach  
and elsewhere I created and started the developement of the subject of
models of SDG adapted (or well adapted as I call them) for the applications to 
classical differential geometry.
            
            [1] Sur les Modeles de la Geometrie Differentielle 
Sinthetyque, "Cahiers de Topologie et Geometrie Differentielle" Vol. XX-3 
(1979).
            [2] Schemas C-inf (amplified version of [3], with detailed 
proofs and many examples), "Prepublications de la Universite de Montreal" 
80-81 edited by  G. Reyes (1980).
            [3] C-inf Schemes, "American Journal of Mathematics", John 
Hopkins University, Vol. 103-4 (1981).
            [4] Open Covers and Infinitary Operations in C-inf-rings, 
"Cahiers de Topologie et Geometrie Differentielle" Vol. XXII-3 (1981).
            [5] Archimedian Local C-inf-rings and Models of SDG (with 
Marta Bunge), "Cahiers de Topologie et Geometrie Differentielle" Vol. 
XXVII-3 (1986).
            [6] Germ representability and Local integration of vector
fields in a well adapted model of SDG,  "Journal of Pure and Applied
Algebra" Vol. 64 (1990).
            
            A) I introduced the notion of well adapted models and 
constructed the first ones.

            B) I started a systematic study of C-infinity rings as such. 
Of course, they were already there, but nothing had been done with them. 
I had to state and prove even such simple facts as that the algebraic 
quotient of a C-infinity ring by an R-algebra ideal had a canonical 
structure of C-infinity ring, and was then the quotient in this category. 

             I introduced the notion of Germ Determined Ideal (or ideal 
of local nature), which was, as such, nowhere to be found in the 
literature, and stated and proved their basic properties. 

            This, I think, is the most important concept in the subject. 
It is the basic definition to start to build upon. It is just the right 
concept needed. Among other things, I first proved it contains all 
finitely generated ideals, and defines the largest possible class of 
C-infinity rings consistent with the nullestelentsatz. This means that 
the ring can be seen as the ring of global sections of a C-infinity 
Scheme. The notion of germ determined ideal also determines the right 
notion of C-infinity local ring (notice that I do not say local 
C-infinity ring), that I then developed. I also developed the relative 
C-infinity version of inverting elements universally, and proved that the 
ring of C-infinity functions defined in an open set U inverts universally 
a function which is non-zero exactly on U (every Euclidean open is 
C-infinity  Zariski). Etc, etc, all the basic structure of the C-infinity 
relative version of algebraic geometry.

            Now, [M-R] write "Although this general notion of C-infinity 
ring does not occur as such in classical analysis and differential 
geometry, the main examples do ...  Given the role of these examples of 
C-infinity rings in the classical literature, it is not surprising that 
although the statements of several of the results in this chapter seem 
new, most of their proofs are either known or easily derivable from known 
ones".
            This is, at the least, misleading, and can also be applied to 
many important concepts in mathematics. Of course, when the new concepts 
are introduced, the examples are already there, and the proof of the 
basic properties is easy.  [M-R] ignore the fact that the important thing 
is to identify explicitly the concept, and to identify the right
statements. And this does not come easily. 
            And I repeat, even if C-infinity rings may have been there,
the concept of germ determined ideal was not, and derived concepts and the
statments of their basic essential properties could not even be there.

            C) With this in hand, I introduced the Topos G of sheaves for 
the open covering topology on the dual of the category of C-infinity 
rings presented by a germ determined ideal, and proved all the basic 
important properties, which many times are the correct relative 
C-infinity versions of corresponding properties in algebraic geometry.

            This topos is the best known model in order to do 
applications of SDG to classical differential geometry, and as such, it 
is the most utilized in practice.
            Many early workers in the subject (J. Penon, O. Bruno, F. 
Gago, Yetter, among others) called this topos "The Dubuc Topos". Even 
Moerdijk and Reyes did so in some preprints, although they changed this 
in the published version. 

            Now, [M-R] write "As far as terminology is concerned, we have 
tried to avoid descriptions of the type "the Moerdijk envelop of the 
Reyes topos", in favor of more informative ones". 
           But the true fact was that the only name that was involved was 
"Dubuc", since no other new name was being utilized at the time (of 
course, things as Kock-Lawvere Axiom or Weil algebras were not aimed by 
this philosophy, and "Moerdijk envelop of the Reyes topos" was an 
invention).
            
           D)  I should mention here that I do not ignore the fact that 
my work is  acknowledged and referenced in the monograph  [M-R], and that 
this can be proved in a court of Law. The matter is much more subtle, and 
nobody can deny the evidence of the following consequence of this 
maneuvering: 
            A consequence of this maneuvering is that my name, as time 
passes, and as young people appear, is less and less associated with a 
subject that I created and developed in large part. Namely, models of SDG 
adapted to classical differential geometry. People talk as if the well 
adapted models were always there, or start referring to them in a way 
that may lead inexperienced (in the subject)  readers  to believe that 
these constructions are "M-R way of doing models", as I quoted at the 
beginning of this note. This is what it is actually happening, and no 
arguing can deny it.

            E) This does not do justice to my work, and does not 
corresponds to the true history of the subject.  As an starting point to 
remedy it, I request all workers that need to use the topos G, to refer 
to it as "Dubuc Topos". After all, it is a long tradition in mathematics 
to associate proper names to important concepts or constructions when it 
is justified, as I believe it is the case here.       

         Eduardo J. Dubuc    Buenos Aires, September 2000.