• Corpus ID: 18264418

Absolute algebra and Segal's Gamma sets

@article{Connes2015AbsoluteAA,
  title={Absolute algebra and Segal's Gamma sets},
  author={Alain Connes and Caterina Consani},
  journal={arXiv: Algebraic Geometry},
  year={2015}
}

• A. Connes, C. Consani
• Published 19 February 2015
• Mathematics
• arXiv: Algebraic Geometry

We show that the basic categorical concept of an s-algebra as derived from the theory of Segal's -sets provides a unied description of several constructions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the ap- proaches using monods, 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 s-algebra to a monad on pointed sets. The notion of an s-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the denition 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.