• 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}
}
• 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…
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## References

SHOWING 1-10 OF 33 REFERENCES
From monoids to hyperstructures: in search of an absolute arithmetic
• Mathematics
• 2010
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
The Arithmetic Site
• Mathematics
• 2014
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
• Mathematics
• 2009
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
• Mathematics
• 2012
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
• Mathematics
• 2015
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.