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\section*{field with one element}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{fun}{}\section*{{F-un}}\label{fun}

\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{BorgerAbsoluteGeometry}{Borger's absolute geometry}\dotfill \pageref*{BorgerAbsoluteGeometry} \linebreak
\noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak
\noindent\hyperlink{AlgebraOverF1}{Algebra over $\mathbb{F}_1$}\dotfill \pageref*{AlgebraOverF1} \linebreak
\noindent\hyperlink{Modules}{Modules/Vector spaces over $\mathbb{F}_1$}\dotfill \pageref*{Modules} \linebreak
\noindent\hyperlink{AlgebraicKTheory}{Algebraic K-theory}\dotfill \pageref*{AlgebraicKTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{expositions}{Expositions}\dotfill \pageref*{expositions} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak
\noindent\hyperlink{relation_to_stable_homotopy_theory}{Relation to stable homotopy theory}\dotfill \pageref*{relation_to_stable_homotopy_theory} \linebreak


\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

Various phenomena in the context of [[algebraic geometry]]/[[arithmetic geometry]] (and particularly in the context of [[algebraic groups]]) over [[finite fields]] $\mathbb{F}_q$ turn out to make perfect sense \emph{as expressions in $q$} when extrapolated to the case $q=1$, and to reflect interesting (combinatorial, representation theoretical\ldots{}) facts, even though, of course, there is no actual [[field]] with a single element (since in a field by definition the elements 1 and 0 are distinct).

Motivated by such observations, [[Jacques Tits]] envisioned in (\hyperlink{Tits57}{Tits 57}) a new kind of geometry adapted to the explanation of these identities. [[Christophe Soulé]] then expanded on Tits' ideas by introducing the notion of \emph{field with one element} and studying its fine arithmetic invariants. While there is no field with a single element in the standard sense of \emph{[[field]]}, the idea is that there is some other object, denoted $\mathbb{F}_1$, such that it does make sense to speak of ``geometry over $\mathbb{F}_1$''. Following the French pronunciation one also writes $F_{un}$ (and is thus led to the inevitable pun).

In the [[relative point of view]] the $S$-schemes are schemes with a morphism of schemes over a base scheme $S$; but every $S$-scheme is a scheme over [[Spec(Z)]]. In \textbf{absolute algebraic geometry} all ``generalized schemes'' should live over $Spec(F_1)$ and $Spec(F_1)$ should live \textbf{below} $Spec(\mathbb{Z})$; this is similar to the fact that the quotient stacks like $[*/G]$ live below the single point $*$ (there is a direct image functor from sheaves on a point to sheaves over $[*/G]$). One of the principal and very bold hopes is that the study of $F_{un}$ should lead to a natural proof of [[Riemann conjecture]] (see also MathOverflow \href{http://mathoverflow.net/questions/69389/riemann-hypothesis-via-absolute-geometry}{here}). It was originally suggested by (\hyperlink{Manin95}{Manin 95}) that the [[Riemann hypothesis]] might be solved by finding an $\mathbb{F}_1$-[[analogy|analogue]] of [[André Weil]]`s proof for the case of [[arithmetic curves]] over the [[finite fields]] $\mathbb{F}_q$.

A first proposal for what a [[variety]] ``over $\mathbb{F}_1$'' ought to be is due to (\hyperlink{Soule}{Soul\'e{} 04}). After that a plethora of further proposals appeared, including (\hyperlink{ConnesConsani08}{Connes-Consani 08}).

\hypertarget{BorgerAbsoluteGeometry}{}\subsection*{{Borger's absolute geometry}}\label{BorgerAbsoluteGeometry}

Maybe an emerging consensus is that the preferred approach is [[Borger's absolute geometry]] (\hyperlink{Borger09}{Borger 09}). Here the structure of a [[Lambda-ring]] on a ring $R$, hence on $Spec(R) \to Spec(\mathbb{Z})$, is interpreted as a collection of lifts of all [[Frobenius morphisms]] and hence as [[descent]] data for descent to $Spec(\mathbb{F}_1)$ (which is defined thereby). This definition yields an [[essential geometric morphism]] of [[gros topos|gros]] [[etale toposes]]

\begin{displaymath}
Et(Spec(\mathbb{Z}))
   \stackrel{\overset{}{\longrightarrow}}{\stackrel{\overset{}{\longleftarrow}}{\underset{}{\longrightarrow}}}
  Et(Spec(\mathbb{F}_1))
  \,,
\end{displaymath}
where on the right the notation is just suggestive, the [[topos]] is a suitable one over [[Lambda-rings]]. Here the middle [[inverse image]] is the [[forgetful functor]] which forgets the Lambda structure, and its [[right adjoint]] [[direct image]] is given by the [[arithmetic jet space]] construction (via the [[ring of Witt vectors]] construction).

This proposal seems to subsume many aspects of other existing proposals (see e.g. \hyperlink{LeBruyn13}{Le Bruyn 13}) and stands out as yielding an ``absolute [[base topos]]'' $Et(Spec(\mathbb{F}_1))$ which is rich and genuinely interesting in its own right.

\hypertarget{function_field_analogy}{}\subsection*{{Function field analogy}}\label{function_field_analogy}

[[!include function field analogy -- table]]

\hypertarget{AlgebraOverF1}{}\subsection*{{Algebra over $\mathbb{F}_1$}}\label{AlgebraOverF1}

\hypertarget{Modules}{}\subsubsection*{{Modules/Vector spaces over $\mathbb{F}_1$}}\label{Modules}

It makes good sense to identify the concept of finite rank [[modules]]/[[finite-dimensional vector spaces]] over the field with one element with that of ([[pointed set|pointed]]) [[finite sets]]

\begin{displaymath}
\mathbb{F}_1 Mod_{fin}
  \;\simeq\;
  Set^{\ast/}_{fin}
\end{displaymath}
and hence the [[symmetric group]] $\Sigma_n$ on $n$ [[elements]] with the [[general linear group]] over $\mathbb{F}_1$:

\begin{displaymath}
GL(n,\mathbb{F}_1)
  \;\simeq\;
  \Sigma_n
\end{displaymath}
(e.g. \hyperlink{Cohn04}{Cohn 04, ``puzzle 1''}, \hyperlink{Durov07}{Durov 07, 2.5.6}, \hyperlink{Snyder07}{Snyder 07})

\hypertarget{AlgebraicKTheory}{}\subsubsection*{{Algebraic K-theory}}\label{AlgebraicKTheory}

With the identification $\mathbb{F}_1 Mod \simeq FinSet^{\ast/}$ from \hyperlink{Modules}{above} it follows that the [[algebraic K-theory]] over $\mathbb{F}_1$ is \emph{[[stable cohomotopy]]}:

\begin{displaymath}
\begin{aligned}
    K \mathbb{F}_1
    & \coloneqq\;
    K(\mathbb{F}_1 Mod)
    \\
    & \simeq\;
    K(FinSet)
    \\
    & \simeq
    \mathbb{S}
  \end{aligned}
  \,.
\end{displaymath}
Here in the second step we used the definition of algebraic K-theory for ordinary [[commutative rings]] as the [[K-theory of a permutative category|K-theory of the permutative category]] of modules (\href{K-theory+of+a+permutative+category#examples#OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules}{this example}), in the second step we used the identification of modules over $\mathbb{F}_1$ with [[pointed set|pointed]] [[finite sets]] from \hyperlink{Modules}{above}, and finally we used the identification of the [[K-theory of a permutative category|K-theory of the permutative category]] of [[finite set]] with the [[sphere spectrum]] (\href{K-theory+of+a+permutative+category#StableCohomotopyIsKTheoryOfFinSet}{this example}), which is the spectrum representing [[stable cohomotopy]], by definition.

The perspective that the [[K-theory]] $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[stable Cohomotopy]] has been highlighted in (\hyperlink{Deitmar06}{Deitmar 06, p. 2}, \hyperlink{Guillot06}{Guillot 06}, \hyperlink{Mahanta17}{Mahanta 17}, \hyperlink{DundasGoodwillieMcCarthy13}{Dundas-Goodwillie-McCarthy 13, II 1.2}, \hyperlink{MoravaSomeBackground}{Morava}, \hyperlink{ConnesConsani16}{Connes-Consani 16}).). Generalized to [[equivariant stable homotopy theory]], the statement that [[equivariant K-theory]] $K_G \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[equivariant stable Cohomotopy]] is discussed in \hyperlink{ChuLorscheidSanthanam10}{Chu-Lorscheid-Santhanam 10, 5.3}.

[[!include Segal completion -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Lambda-ring]],


\item [[blue scheme]]


\item [[tropical geometry]].



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

After the very first observations by Tits, pioneers were [[Christophe Soulé]] and Kapranov and Smirnov. More recently there are extensive works by [[Alain Connes]] and Katia Consani, [[Nikolai Durov]], [[James Borger]] and [[Oliver Lorscheid]].

\hypertarget{expositions}{}\subsubsection*{{Expositions}}\label{expositions}

\begin{itemize}%
\item Henry Cohn, \emph{Projective geometry over $\mathbb{F}_1$ and the Gaussian binomial coefficients}, American Mathematical Monthly 111 (2004), 487-495 (\href{https://arxiv.org/abs/math/0407093}{arXiv:math/0407093})


\item [[Lieven Le Bruyn]], \emph{Looking for $F_{un}$}, \href{http://www.neverendingbooks.org/looking-for-f_un}{blog}


\item [[Noah Snyder]], \emph{\href{https://sbseminar.wordpress.com/2007/08/14/the-field-with-one-element/}{The field with one element}}, 2007


\item Javier L\'o{}pez Pe\~n{}a, [[Oliver Lorscheid]], \emph{Mapping $F_1$-land:An overview of geometries over the field with one element}, \href{http://arxiv.org/abs/0909.0069}{arXiv/0909.0069}


\item [[John Baez]], \emph{This Week's Finds 259} (\href{http://math.ucr.edu/home/baez/week259.html}{html} \href{http://golem.ph.utexas.edu/category/2007/12/this_weeks_finds_in_mathematic_19.html}{blog})


\item [[Alain Connes]], \emph{Fun with $\mathbf{F}_1$}, 5 min. \href{http://www.dailymotion.com/video/x6xe0g_article-alain-connes_tech}{video}


\item [[Lieven Le Bruyn]], \emph{The field with one element}, seminar notes 2008 (\href{http://win.ua.ac.be/~lebruyn/b2hd-LeBruyn2008d.html}{web})


\item [[Oliver Lorscheid]], \emph{Lectures on $\mathbb{F}_1$}, 2014 (\href{http://www.impa.br/~lorschei/Lectures_on_F1.pdf}{pdf})


\item [[Oliver Lorscheid]], \emph{$\mathbb{F}_1$ for everyone}, 2018 (\href{https://arxiv.org/abs/1801.05337}{arXiv:1801.05337})



\end{itemize}
\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

\begin{itemize}%
\item [[Jacques Tits]], \emph{Sur les analogues algebriques des groupes semi-simples complexes}. In Colloque d'algebre superieure, tenu a Bruxelles du 19 au 22 decembre 1956, Centre Belge de Recherches Mathematiques, pages 261-289. Etablissements Ceuterick, Louvain, 1957.


\item [[Christophe Soulé]], \emph{Les varietes sur le corps a un element} Mosc. Math. J., 4(1):217-244, 312, 2004 (\href{http://www.ams.org/distribution/mmj/vol4-1-2004/soule.pdf}{pdf})


\item [[Yuri Manin]], \emph{Lectures on zeta functions and motives (according to Deninger and Kurokawa)} Asterisque, (228):4, 121-163, 1995. Columbia University Number Theory Seminar (\href{http://streams1.nts.jhu.edu/mathematics/ncgdocs/manin-zeta-absmotives.pdf}{pdf})


\item [[Bertrand Toen]], [[Michel Vaquie]], \emph{Under Spec Z} (\href{http://arxiv.org/abs/math/0509684}{arXiv:math/0509684})



\end{itemize}
Around (0.4.24.2) in

\begin{itemize}%
\item [[Nikolai Durov]], \emph{New Approach to Arakelov Geometry} (\href{http://arxiv.org/abs/0704.2030}{arXiv:0704.2030})

\end{itemize}
the algebraic structure of $\mathbb{F}_1$ is regarded as being the [[maybe monad]], hence [[modules]] over $\mathbb{F}_1$ are defined to be [[algebra over a monad|monad-algebras]] over the [[maybe monad]], hence [[pointed sets]].

Other approaches include

\begin{itemize}%
\item [[Alain Connes]], [[Caterina Consani]], [[Matilde Marcolli]], \emph{Fun with $\mathbf{F}_1$}, \href{http://arxiv.org/abs/0806.2401}{arxiv/0806.2401}


\item [[Yuri Manin]], \emph{Cyclotomy and analytic geometry over $F_1$}, \href{http://arxiv.org/abs/0809.1564}{arxiv/0809.1564}


\item [[Alain Connes]], [[Caterina Consani]], \emph{On the notion of geometry over $\F_1$}, \href{http://arxiv.org/abs/0809.2926}{arxiv/0809.2926};

\emph{Schemes over $\F_1$ and zeta functions}, \href{http://arxiv.org/abs/0903.2024}{arxiv/0903.2024}; \emph{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, \href{http://arxiv.org/abs/0911.3537}{arxiv/0911.3537};

\emph{From monoids to hyperstructures: in search of an absolute arithmetic}, \href{http://arxiv.org/abs/1006.4810}{arxiv/1006.4810};

\emph{On the arithmetic of the BC-system}, \href{http://arxiv.org/abs/1103.4672}{arxiv/1103.4672}; \emph{Projective geometry in characteristic one and the epicyclic category}, \href{http://arxiv.org/abs/1309.0406}{arxiv/1309.0406}


\item [[M. Marcolli]], [[Ryan Thorngren]], \emph{Thermodynamical semirings}, \href{https://arxiv.org/abs/1108.2874}{arXiv/1108.2874}


\item Bora Yalkinoglu, \emph{On Endomotives, Lambda-rings and Bost-Connes systems}, With an appendix by Sergey Neshveyev, \href{http://arxiv.org/abs/1105.5022}{arxiv/1105.5022}



\end{itemize}
The approach in terms of [[Lambda-rings]] due to

\begin{itemize}%
\item [[James Borger]], \emph{Lambda-rings and the field with one element} (\href{http://arxiv.org/abs/0906.3146}{arXiv/0906.3146})

\end{itemize}
with details in

\begin{itemize}%
\item [[James Borger]], \emph{The basic geometry of Witt vectors, I: The affine case} (\href{http://arxiv.org/abs/0801.1691}{arXiv:0801.1691})


\item [[James Borger]], \emph{The basic geometry of Witt vectors, II: Spaces} (\href{http://arxiv.org/abs/1006.0092}{arXiv:1006.0092})



\end{itemize}
More discussion relating to this includes

\begin{itemize}%
\item [[Lieven Le Bruyn]], \emph{Absolute geometry and the Habiro topology} (\href{http://arxiv.org/abs/1304.6532}{arXiv:1304.6532})

\end{itemize}
\hypertarget{relation_to_stable_homotopy_theory}{}\subsubsection*{{Relation to stable homotopy theory}}\label{relation_to_stable_homotopy_theory}

The interpretation of [[stable cohomotopy]] as [[algebraic K-theory]] over $\mathbb{F}_1$ is amplified in the following articles:

\begin{itemize}%
\item [[Anton Deitmar]], \emph{Remarks on zeta functions and K-theory over $\mathbb{F}_1$} (\href{https://arxiv.org/abs/math/0605429}{arXiv:math/0605429})


\item [[Pierre Guillot]], \emph{Adams operations in cohomotopy} (\href{https://arxiv.org/abs/math/0612327}{arXiv:0612327})


\item [[Jack Morava]], Rekha Santhanam, \emph{Power operations and absolute geometry}, 2012 (\href{http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf}{pdf})


\item [[Snigdhayan Mahanta]], \emph{G-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem}, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 (\href{https://arxiv.org/abs/1110.6001}{arXiv:1110.6001})


\item John D. Berman, p. 92 of \emph{Categorified algebra and equivariant homotopy theory}, PhD thesis 2018 (\href{http://www.people.virginia.edu/~jdb8pc/Thesis.pdf}{pdf})


\item Chenghao Chu, [[Oliver Lorscheid]], Rekha Santhanam, \emph{Sheaves and K-theory for $\mathbb{F}_1$-schemes}, Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2239-2286 (\href{https://arxiv.org/abs/1010.2896}{arxiv:1010.2896})



\end{itemize}
see also

see also

\begin{itemize}%
\item [[Jack Morava]], \emph{Some background on Manin's theorem $K(\mathbb{F}_1) \sim \mathbb{S}$} (\href{http://www.alainconnes.org/docs/Morava.pdf}{pdf}, [[MoravaSomeBackground.pdf:file]])


\item [[Alain Connes]], [[Caterina Consani]], \emph{Absolute algebra and Segal's Gamma sets}, Journal of Number Theory 162 (2016): 518-551 (\href{https://arxiv.org/abs/1502.05585}{arXiv:1502.05585})


\item John D. Berman, p. 92 of \emph{Categorified algebra and equivariant homotopy theory}, PhD thesis 2018 (\href{http://www.people.virginia.edu/~jdb8pc/Thesis.pdf}{pdf})



\end{itemize}
[[!redirects field of one element]] [[!redirects Fun]] [[!redirects absolute geometry]] [[!redirects absolute algebraic geometry]] [[!redirects absolute ground field]] [[!redirects F1]]



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