Query about SDG (Synthetic Differential Geometry)

Question

(Edited 10/17/17): With the hope of obtaining informed responses on the following intriguing remark of Marta Bunge on the status of Synthetic Differential Geometry, I have added a third question to the original two and expanded Bunge's quote to provide further context.

In her "A Personal tribute to Bill Lawvere", presented at Union College in January 2013, Marta Bunge made the following observation about synthetic differential geometry (SDG):

The basic idea of Synthetic Differential Geometry, in the form of the Kock-Lawvere axiom, requires, for a topos $\mathcal{E}$ with a ring object $\textrm{R}$ in it, that the subobject $\textrm{D}$ of $\textrm{R}$, consisting of those elements of square zero, be tiny and representing of tangent vectors at $0$ of arrows from $\textrm{R}$ to $\textrm{R}$. During the period 1981-88, I devoted myself almost totally to SDG, involving students and collaborators (Murray Heggie, Patrice Sawyer, Eduardo Dubuc, Felipe Gago) and participating in the workshops organized by Anders Kock at Aarhus, as well as in related special meetings. Lawvere’s intuition of the role of atoms (or “tiny objects”) in developing a simple form of Analysis going back to the ideas of Newton and Leibniz, and in the same spirit as in the work of André Weil, was both simple and attractive. In my work with my student Felipe Gago on a synthetic theory of smooth mappings, we used two additional axioms (Bunge-Dubuc 1987) to SDG, to wit, the representability of germs of smooth mappings by the sub object $\Delta = \neg \neg \{0\}$ of $\textrm{R}$, required to be tiny, and the existence and uniqueness of solutions of ordinary differential equations. However, no well adapted model of SDG is known at present to satisfy both of these axioms. This open problem is, in my view, pivotal for further progress in this fascinating area, which includes a synthetic proof of Mather’s theorem on the equivalence of locally stable and infinitesimally stable germs of smooth mappings (Bunge-Gago 1988), as well as Morse theory, developed synthetically in the thesis work of Felipe Gago at McGill.

I have three questions regarding this remark.

(i) Is the problem of the existence of a well adapted model of SDG that satisfies the above-stated two additional axioms still open?

(ii) Assuming it is, how widely shared is her view on this matter in the SDG research community? That is, how widely is it held by members of the community that further progress in SDG is contingent on the existence of such a well adapted model?

(iii) (New Question): Assuming the two additional assumptions are natural, which they appear to be, are there any cogent arguments opposed to her view?

Answer 1

In a paper by Marta Bunge and Eduardo Dubuc. "Local concepts in SDG and germ representability" (1987) certain axioms were laid down towards a synthetic theory of differential topology based on logical infinitesimal notions given by Jacques Penon in his 1985 Universite Paris VII thesis.

One of them was Postulate WAII on $\Delta$-integration of vector fields, where $\Delta = \neg \neg \{0\}$ is a subobject of $R$ (the ring of line type in the Kock-Lawvere axioms) that represents germs of mappings from $R$ to $R$ by one of the basic axioms of what was to become SDT. As shown therein, this postulate is equivalent to the existence and uniqueness of (local) solutions to ODE within SDG.

On the basis of Postulate WAII, along with further axioms for a theory now called SDT (Synthetic Differential Topology) an extension of SDG, Marta Bunge and her McGill University student Felipe Gago in "Synthetic aspects of $C^{\infty}$-mappings II : Mather's theorem for infinitesimally represented germs" (1988) proved the theorem of the title. Felipe Gago in "Morse theory for infinitesimally represented germs" (1988) developed Morse theory which was part of his 1988 McGill University thesis. In her 1999 thesis, Ana Maria San Luis, a student of Felipe Gago at Sgo de Compostela, gave an alternative proof of Mather's Theorem (without the so-called Preparation Theorem) still using Postulate WAII.

Subsequently, Eduardo Dubuc, in "Germ representability and local integration of vector fields in a well adapted model of SDG" (1980) gave proofs of the validity of the basic axiom of germ representability as well as of Postulate WAII both in the topos $(G, R)$ with $G$ the topos of sheaves on the opposite of the category $B$ of finitely generated $C^{\infty}$-rings determined by a local (or germ determined) ideal, known as the Dubuc topos. Unfortunately, an error in the validity of the uniqueness part of the proof of Postulate WAII in the Dubuc topos $G$ given therein was found by Michael Makkai in discussions with Gonzalo Reyes.

This meant therefore that there was at that time no known well adapted model of all of the axioms and postulates used in the work of Bunge-Gago-San Luis on a synthetic theory of smooth mappings and their singularities. Hence the remarks made by myself concerning the need to find a suitable well adapted model of SDT to validate our work. At the time this was indeed an open question but it is no longer one. As shown by myself at the Octoberfest, Ottawa, October 31-November 1, 2015, in a talk "Synthetic Theory of Stable Mappings and their Singularities" (whose slides have been posted in my Research Gate page), the uniqueness part of Postulate WAII (called Postulate VIII therein) is in fact valid in the Dubuc topos $G$, which is then a well adapted model of SDT (as well as of SDG).

This result is now included in Chapter 12 ("$G$ as a WAM of SDT") of a forthcoming book by Marta Bunge, Felipe Gago and Ana San Luis, "Synthetic Differential Topology", Cambridge University Press, to appear in 2018. In Chapter 12, proofs of the validity of all (general and special) axioms and postulates used in our work are given with references to their various sources. Further developments of differential topology might need additional axioms and postulates, but it seems reasonable now (as it was so before the error was found!) to expect that those too will be shown valid in $(G, R)$.

An interesting characterization of well adapted models of SDG (and so also of SDT) involving the Dedekind reals in a topos was given in a 1980 paper by Marta Bunge and Eduardo Dubuc "Archimedian local $C^{\infty}$-rings and models of SDG".