Field With One Element - Properties

Properties

F1 is believed to have the following properties.

  • Finite sets are both affine spaces and projective spaces over F1.
  • Pointed sets are vector spaces over F1.
  • The finite fields Fq are quantum deformations of F1, where q is the deformation.
  • Weyl groups are simple algebraic groups over F1:
    Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is the semisimple algebraic group over F1.
  • Spec Z is a curve over F1.
  • Groups are Hopf algebras over F1. More generally, anything defined purely in terms of diagrams of algebraic objects should have an F1-analog in the category of sets.
  • Group actions on sets are projective representations of G over F1, and in this way, G is the group Hopf algebra F1.
  • Toric varieties determine F1-varieties. In some descriptions of F1-geometry the converse is also true, in the sense that the extension of scalars of F1-varieties to Z are toric Whilst other approaches to F1-geometry admit wider classes of examples, toric varieties appear to lie at the very heart of the theory.
  • The zeta function of PN over F1 should be ζ(s) = s(s − 1)···(s − N).
  • The mth K-group of F1 should be the mth stable homotopy group of the sphere spectrum.

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