# The Universal Property of Projective Space

Since the theories of affine and projective geometry are specified by certain axioms, we can consider the category $\mathcal{A}$ and $\mathcal{P}$ of affine and projective geometries, whose morphisms are maps preserving colinearity. Given an affine geometry $G$, there is a natural way to extend the geometry to a projective geometry $G_\infty$ by adding a single `line at infinity' on which all parallel lines intersect. The association $G \mapsto G_\infty$ is trivially a functor, because a map $G \to H$ induces a map $G_\infty \to H_\infty$, and these maps behave properly under composition. My question is whether there is a universal property which uniquely describes the projective extension of an affine geometry up to isomorphism, like we see for the universal properties of the free group over a set, or the Stone-Cech compactification.

• I am not very well-versed with this, but I was under the impression that this is what the fundamental theorem of projective geometry did, in one formulation or another. Except that I thought the upshot was that the equivalence was not exactly isomorphism but semilinear isomorphism. Commented May 9, 2017 at 20:14
• What I've written is not an answer of course because I'm not very practiced with it. Maybe you can tell me why it isn't sufficient/directly applicable. Commented May 9, 2017 at 20:15
• The fundamental theorem of projective geometry only holds over projective geometries of the form $PK^2$, whereas there are projective geometries which do not have to be isomorphic to a geometry of this form (The non Desarguian geometries). Furthermore, even if we restricted our categories to geometries isomorphic to $K^2$ and $PK^2$ (those geometries satisfying Desargues + Pappus' theorem), I still don't see how the fundamental theorem characterizes a universal property of the embedding $K^2 \to PK^2$. Perhaps you could elaborate? Commented May 9, 2017 at 21:36
• Ah yeah, I was assuming nice planes that were at least Desarguesian. Projective planes in general are pretty ambitious! I think that the fundamental theorem is still valid for any Desarguesian plane, even non-Pappian ones, and also for higher dimensions, but you're right, perhaps it is not talking about a universal property. Commented May 9, 2017 at 23:50
• I think, with the category of affine planes with nonsingular affine transformations, and the category of projective planes with a distinguished line and nonsingular projective transformations, the connection is actually an equivalence of categories, isn't it? Doesn't the forgetful functor that "forgets" the distinguished ideal line act as an inverse to the functor you mentioned? Commented May 10, 2017 at 0:52