Mapping F_1-land:An overview of geometries over the field with one element

TL;DR

This paper surveys various approaches to F_1-geometry, compares their frameworks, and constructs functors linking these theories, culminating in insights into Chevalley groups over F_1.

Contribution

It provides a comprehensive overview of existing F_1-geometries and establishes a unifying diagram connecting different theories with new insights.

Findings

Unified framework of F_1-geometries via functors

Connections between diverse F_1-theories established

Realization of Tits' idea about Chevalley groups over F_1

Abstract

This paper gives an overview of the various approaches towards F_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar's F_1-schemes, To\"en and Vaqui\'e's F_1-schemes, Haran's F-schemes, Durov's generalized schemes, Soul\'e's varieties over F_1 as well as his and Connes-Consani's variations of this theory, Connes and Consani's F_1-schemes, the author's torified varieties and Borger's Lambda-schemes. In a second part, we will tie up these different theories by describing functors between the different F_1-geometries, which partly rely on the work of others, partly describe work in progress and partly gain new insights in the field. This leads to a commutative diagram of F_1-geometries and functors between them that connects all the reviewed theories. We conclude the paper by reviewing the second author's constructions that lead to realization of Tits' idea about Chevalley groups over F_1.