Universal projective geometry via topos theory

Abstract

The idea of this article is that linear algebra and projective geometry over a local commutative ring is equivalent to intuitionistic linear algebra and intuitionistic pure projective geometry over a field (at least in so far as caherenr sentences are concerned; this term will be explained). By pure projective geometry is meant a formulation of synthetic geometry in terms of the predicates “incidence” and “equality” alone, in contrast to formulations which include a special apartncss predicate. Heyting’s intuitionistic projective geometry, for instance, has such an apartness predicate o, and it is also necessary to have such w if one wants to formulate a reasonable synthetic theory for the projective geometry over a local ring. There are probably other and better reasons for looking for the “projective geometry over a local ring” than the one which gave rise to the present research: Study’s transfer principle. It says that whatever is true in plane projective geometry over the ring of dual numbers D = R[E] (with E’ = 0), can be reinterpreted as a theorem about the set of lines in euclidean 3-spL.:e. see e.g. [C;). This set (which is a 4-dimensional manifold) is thereby made into a Hjelmslev plune; Hjelmslev used this fact. Of course, the value of the transfer principle is that one gets, or hopes to get, theorems for the projective plane over D ;~y “analogy” with the well-known projective geometry over R. We are stud;% ig the meta-mathematics of that “analogy”.