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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….

I have two questions regarding this remark.

(i) Is the problem of the existence of a well adapted model of SDG that satisfies Dubuc and her 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 a well adapted model satisfying the two additional conditions due to her and Dubuc?

(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?

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 $E$ with a ring object $R$ in it, that the subobject $D$ of $R$, consisting of those elements of square zero, be tiny and representing of tangent vectors at 0 of arrows from $R$ to $R$. 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 $R$, 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….

I have two questions regarding this remark.

(i) Is the problem of the existence of a well adapted model of SDG that satisfies Dubuc and her 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 a well adapted model satisfying the two additional conditions due to her and Dubuc?

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….

I have two questions regarding this remark.

(i) Is the problem of the existence of a well adapted model of SDG that satisfies Dubuc and her 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 a well adapted model satisfying the two additional conditions due to her and Dubuc?

A few years ago, in her "A Personal tribute to Bill Lawvere", 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 $E$ with a ring object $R$ in it, that the subobject $D$ of $R$, consisting of those elements of square zero, be tiny and representing of tangent vectors at 0 of arrows from $R$ to $R$. 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 $R$, 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….

I have two questions regarding this remark.

(i) Is the problem of the existence of a well adapted model of SDG that satisfies Dubuc and her 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 a well adapted model satisfying the two additional condition due to her and Dubuc?

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 $E$ with a ring object $R$ in it, that the subobject $D$ of $R$, consisting of those elements of square zero, be tiny and representing of tangent vectors at 0 of arrows from $R$ to $R$. 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 $R$, 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….

I have two questions regarding this remark.

(i) Is the problem of the existence of a well adapted model of SDG that satisfies Dubuc and her 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 a well adapted model satisfying the two additional conditions due to her and Dubuc?