The Arithmetic Site

The Arithmetic Site is the presheaf topos PSh(NPos) together with its commutative-semiring-valued structure sheaf O = NBar.

This is a ringed topos of characteristic 1: the structure sheaf has idempotent addition, placing the construction in the world of tropical algebra. Connes and Consani describe it as the "algebraic geometric incarnation" of the non-commutative geometric approach to the Riemann hypothesis.

def ArithmeticSite.arithmeticSite : CategoryTheory.Functor BNplusᵒᵖ CommSemiRingCat

The semiring-valued presheaf that supplies the structure sheaf of the Arithmetic Site. Its domain fixes the ambient topos to PresheafTopos, and forgetting its algebraic structure gives structureSheaf.

This is an honest package of the additional structure currently formalized; it is not claimed to be a general-purpose bundled notion of ringed topos.

Equations

• ArithmeticSite.arithmeticSite = ArithmeticSite.structureSemiring

theorem ArithmeticSite.arithmeticSite_forget : arithmeticSite.comp (CategoryTheory.forget CommSemiRingCat) = structureSheaf