Abstract Projective Lines

TL;DR

This paper introduces an abstract framework for projective lines over fields using first-order structures and groupoids, leading to a stack of projective lines with a natural notion of bundle and projectivity.

Contribution

It formalizes the concept of projective lines via first-order structures and groupoids, providing a new categorical perspective and a stack-based approach.

Findings

Defines an abstract projective line as a set with a first-order structure

Introduces projectivities as structure-preserving bijections

Develops a stack of bundles of projective lines

Abstract

We describe a notion of (abstract) projective line over a field as a set equipped with a certain first order structure, and a projectivity between projective lines as a bijection preserving this structure. The structure in question is that of a groupoid, with certain properties. This leads to a natural notion of bundle of projective lines, forming a stack.