Geometric algebra of projective lines

TL;DR

This paper explores how the structure of the projective line over a field, viewed as a groupoid, can be used to reconstruct the underlying field, revealing deep connections between geometry and algebra.

Contribution

It introduces a framework for reconstructing fields from the groupoid structure of the projective line, advancing understanding of geometric-algebraic correspondences.

Findings

The projective line over a field forms a groupoid with specific properties.

The field can be reconstructed from the groupoid structure under certain conditions.

The work clarifies the relationship between geometric structures and their algebraic origins.

Abstract

The projective line over a field carries structure of a groupoid with a certain correspondence between objects and arrows. We discuss to what extent the field can be reconstructed from the groupoid.