symmetric monoidal (∞,1)-category of spectra
Various identities coming from algebraic geometry (and particularly algebraic groups) over finite fields turn out to make perfect sense as expressions in when extrapolated to the case , and to reflect interesting (combinatorial, representation theoretical…) facts, even though, of course, there is no actual field with a single element.
Motivated by such observations, Jacques Tits envisioned a new kind of geometry adapted to the explanation of these identities. Christophe Soulé then expanded on Tits’ ideas by introducing the notion of field with one element and studying its fine arithmetic invariants. While there is no field with a single element in the standard sense of field, the idea is that there is some other object, denoted , such that it makes sense to speak of “geometry over ”. Following the French pronunciation one also writes (and is thus led to the inevitable pun).
In the relative point of view the -schemes are schemes with a morphism of schemes over a base scheme ; but every -scheme is a scheme over . In absolute algebraic geometry all “generalized schemes” should live over and should live below ; this is similar to the fact that the quotient stacks like live below the single point (there is a direct image functor from sheaves on a point to sheaves over . One of the principal and very bold hopes is that the study of should lead to a natural proof of Riemann conjecture; namely some attractive strategies to such a proof already exist (see MathOverflow here).
After the very first observations by Tits, pioneers were Soulé and Kapranov and Smirnov. More recently there are extensive works by Alain Connes and Katia Consani, Nikolai Durov, James Borger and Oliver Lorscheid.
A survey of the various competing theories is
Javier López Peña, Oliver Lorscheid, Mapping -land:An overview of geometries over the field with one element, arXiv/0909.0069
Alain Connes, Caterina Consani?, On the notion of geometry over , arxiv/0809.2926; Schemes over and zeta functions, arxiv/0903.2024; Characteristic one, entropy and the absolute point, in: Noncommutative Geometry, Arithmetic, and Related Topics, 21st Meeting of the Japan-U.S. Math. Inst., Baltimore 2009, JHUP (2012), pp. 75–139, arxiv/0911.3537; From monoids to hyperstructures: in search of an absolute arithmetic, arxiv/1006.4810; On the arithmetic of the BC-system, arxiv/1103.4672; Projective geometry in characteristic one and the epicyclic category, arxiv/1309.0406
Bora Yalkinoglu, On Endomotives, Lambda-rings and Bost-Connes systems, With an appendix by Sergey Neshveyev, arxiv/1105.5022
An approach in terms of Lambda-rings is