Introduction

The question of models should be viewed from two angles. One is the purely synthetic: we consider some property or structure which we out of experience think the geometric line has, and we experiment with the property to see whether it is logically consistent with other desirable or true properties. To this end we construct models (which are often of quite algebraic character).

The other angle from which we view the question of models is to compare the synthetic theory with the analytic, for the benefit of both. The mutual benefit may for instance take form of a definite comparison theorem to the effect that properties proved or constructions performed synthetically hold or exist in the analytic theory, too, or vice versa. The models that give rise to the comparisons usually contain the category of smooth manifolds as a full subcategory, and are called well-adapted models (= well adapted for the comparison).

In §§1–2 we treat algebraic models, and in the rest the well-adapted ones. The latter are treated by a quite algebraic method, namely using the “algebraic theory of smooth functions”.