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

Understand

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.