Absolute algebra and Segal's Γ-rings : Au dessous de Spec(Z)‾

TextWe show that the basic categorical concept of an SS-algebra as derived from the theory of Segal's Γ-sets provides a unifying description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoïds, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an SS-algebra to a monad on pointed sets. The notion of an SS-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homology. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories.VideoFor a video summary of this paper, please visit https://youtu.be/BJ4Zsc4Nv7E.