Loose ends

There are several developments, related to the content of the present book, which were not treated in it, due to lack of time or space, or lack of comprehension on my side. I would have liked to include something about:

1. Formulation and application of a synthetic inverse function theorem. A formulation has been given by Penon [64]: he utilizes 1) that an inverse function theorem expresses that a germ is invertible if its 1-jet is invertible 2) a synthetic formulation of the germ notion is implied by his Theorem (Theorem III.10.3) (and, of course, a synthetic formulation of the jet-notion is implied in the foundations of synthetic differential geometry).

In particular, a synthetic version of the Preimage theorem (as quoted in III §3) is still lacking.

2. Differential equations. A correct formulation of the classical existence and uniqueness theorems for flows for vector fields is still lacking. The problem is that solutions (flows) only should exist locally, (cf. e.g. Exercise I.8.7). Maybe the Penon germ notion implied in Theorem III.10.3 will provide a formulation, which is true in some of the models, and strong enough to carry come synthetic theory.

3. Calculus of variations. The problems and perspectives are related to those mentioned for differential equations. However, a gros topos designed for this purpose, and for the purpose of considering differential forms as quantities, exists implicitly in work of K.T. Chen, as Lawvere has pointed out.