Projective lines as groupoids with projection structure
type: Item
instance of: scholarly article
zbMATH Open document ID: 1308.18002
author: Anders Kock
publication date: 2014-08-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.5357
full work available at URL: 29-13abs.html
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 18D99
Mathematics Subject Classification ID: 51A25
zbMATH DE Number: 6334031
zbMATH Keywords: projective geometry
zbMATH Keywords: groupoid
zbMATH Keywords: projection structure
zbMATH Keywords: cross-ratio
zbMATH Keywords: bi-rapport
zbMATH Keywords: tri-rapport
MaRDI profile type: Publication
arXiv ID: 1310.5357
arXiv classification: math.CT
title: Projective lines as groupoids with projection structure
published in: Theory and Applications of Categories
review text: The author defines a projection structure on a groupoid \(\mathbb{L}\) as a set of bijections between the set of arrows \(A \to B\) and \(\mathbb{L}\setminus\{A,B\}\) for any two distinct points \(A\), \(B \in \mathbb{L}\). The projective line \(P(V)\) over a two-dimensional \(k\)-vector space \(V = k^2\) is naturally equipped with such a projection structure: The restriction of the vector space isomorphism taking the subspace \(A\) to the subspace \(B\) is a projection parallel to subspace \(C \neq A, B\). The aim of this article is to recover the vector space from an axiomatically defined projection structure. To this end, the author introduces four axioms: Axioms 1 and 2 concern commutativity properties of arrows, Axioms 3 and 4 are concern permutation properties of bi-rapports (cross-ratios) and tri-rapports.NEWLINENEWLINEThe scalars are defined as elements of the abstract vertex group (all vertex groups are isomorphic). Hence, bi- and tri-rapports are compositions of certain arrows. The author relates bi- and tri-rapports, develops some basic arithmetic properties of the scalar group, proves a fundamental theorem and, under some mild additional assumptions that make the scalar group a field, identify the groupoid with the projective line over its scalar field.NEWLINENEWLINEThe article closes with the suggestion to develop a weaker axiomatic theory that also covers projective lines over rings.
reviewed by: Hans-Peter Schröcker
description: scientific article; zbMATH DE number 6334031