Hyper-Operations and Extension of Scalars from ${\mathbb{F}}_1$ to $\mathbb{Z}$

Abstract

The additive structure of ${\mathbb{F}}_1$-modules (in the sense of Segal's $\Gamma$-sets) differs fundamentally from that of abelian groups: addition is encoded through a family of $n$-ary hyper-operations that are multivalued and do not satisfy classical associativity. We establish a \emph{law of generalized associativity} showing that, despite this failure of strict associativity, all $n$-ary sums are controlled by successive binary operations. This enables us to construct an extension of scalars functor $-{\otimes }_{{\mathbb{F}}_1}\mathbb{Z}:{\mathbb{F}}_1\mathbf{Mod}\to \mathbf{Ab}$ that universally strictifies the hyper-additive structure of ${\mathbb{F}}_1$-modules into classical abelian group addition. We prove this functor is left adjoint to the Eilenberg-MacLane functor $H:\mathbf{Ab}\to {\mathbb{F}}_1\mathbf{Mod}$. Extending to the multiplicative setting, we obtain an adjunction $-{\otimes }_{{\mathbb{F}}_1}\mathbb{Z}:{\mathbb{F}}_1\mathbf{Alg}\leftrightarrows \mathbf{CRing}:H$ between commutative ${\mathbb{F}}_1$-algebras and commutative rings. This recovers Deitmar's monoid ring construction for spherical monoid algebras and provides a base change mechanism needed for absolute algebraic geometry.

28 pages