• 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}
}
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… 

Figures from this paper

Algebraic Geometry Over Hyperrings
SpecZ and the Gromov norm
We define the homology of a simplicial set with coefficients in a Segal’s Γ-set (s-module). We show the relevance of this new homology with values in s-modules by proving that taking as coefficients
BC-system, absolute cyclotomy and the quantized calculus
We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the ”zeta sector” of the latter space as the Scaling Site. The new
Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives
We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These
On Noetherian schemes over (C,⊗,1) and the category of quasi-coherent sheaves
Let (C,⊗,1) be an abelian symmetric monoidal category satisfying certain conditions and let X be a scheme over (C,⊗,1) in the sense of Toen and Vaquie. In this paper we show that when X is
Riemann-Roch for Spec Z
We prove a Riemann-Roch formula for Arakelov divisors on Spec Z equating the integer valued Euler characteristic with a simple modifica-tion of the traditional expression ( i.e. the degree of the
On Noetherian schemes over (\mathcal{C},\otimes,1)$ and the category of quasi-coherent sheaves
Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of To\"en and Vaqui\'{e}. In
$\overline{Spec\mathbb Z}$ and the Gromov norm
We define the homology of a simplicial set with coefficients in a Segal's $\Gamma$-set ($\mathbf S$-module). We show the relevance of this new homology with values in $\mathbf S$-modules by proving

References

SHOWING 1-10 OF 33 REFERENCES
From monoids to hyperstructures: in search of an absolute arithmetic
We show that the trace formula interpretation of the explicit formulas expresses the counting functionN.q/ of the hypothetical curveC associated to the Riemann zeta function, as an intersection
Simplicial functors and stable homotopy theory
The problem of constructing a nice smash product of spectra is an old and well-known problem of algebraic topology. This problem has come to mean the following: Find a model category, which is
New Approach to Arakelov Geometry
This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct
Characteristic one, entropy and the absolute point
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion
The universal thickening of the eld of real numbers
We dene the universal 1-adic thickening of the eld of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion
Universal Thickening of the Field of Real Numbers
We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process.
...
...