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.