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 = \not not {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".

Marta Bunge November 25, 2017

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".