Morphisms of Projective Geometries

Abstract

Since a morphism from a projective geometry G ₁ to another one G ₂ is a map that is defined on certain, but in general not on all points of G ₁, we begin the chapter with some generalities on the so-called partial maps. A partial map from a set X to a set Y, noted f: X - -→ Y,is a map f from a subset of X, called the domain of f,to Y. The points of X which are not in the domain of f form the so-called kernel of f. The sets together with the partial maps form a category Par. In the first section we give some elementary basic facts concerning this category. One can avoid partial maps by working with the equivalent category Set* of pointed sets. This is sometimes quite useful, but it involves that one has to add to the points of every projective geometry an additional point which can be considered as the zero element of the lattice of subspaces, but has no direct geometric interpretation.