A constructive approach to affine and projective planes

Abstract

In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local rings instead of fields. We start by constructing projective and affine planes over local rings and establishing forms of Desargues' Theorem and Pappus' Theorem which hold for these. From this analysis we derive coherent theories of projective and affine planes. The great Greek mathematicians of the classical period used geometry as the basis for their theory of quantities. The modern version of this idea is the reconstruction of algebra from geometry. We show how we can construct a local ring whenever we are given an affine or a projective plane. This enables us to describe the classifying toposes of our theories of affine and projective planes as extensions of the Zariski topos by certain group actions. Through these descriptions of the classifying toposes, the links between the theories of local rings, affine and projective planes become clear. In particular, the geometric morphisms between these classifying toposes are all induced by group homomorphisms even though they demonstrate complicated constructions in geometry. In this thesis, we also prove results in topos theory which are applied to these geometric morphisms to give Morita equivalences between some further theories.