A few years ago, in her ["A Personal tribute to Bill Lawvere"](http://www.math.union.edu/~niefiels/13conference/Web/Tributes/bunge.pdf),
Marta Bunge made the following observation about smooth 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 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?