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\begin{document}

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\section*{Schanuel topos}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

[[!redirects Myhill-Schanuel topos]] [[!redirects Schanuel Topos]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_concepts}{Related Concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{Schanuel topos} (also called the \emph{Myhill-Schanuel topos}) is the [[Grothendieck topos]] of combinatorial functors. It plays an important role in [[computer science]] in the theory of name-binding calculi and in [[William Lawvere]]`s approach to petit toposes. It can be viewed as a categorical variant of the [[Fraenkel-Mostowski model]] of [[set theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The \emph{Schanuel topos} is the [[sheaf topos]] $Sh(FinSet^{op}_{mono},J)$ where $FinSet^{op}_{mono}$ is the opposite of the category of finite sets and monomorphisms and the coverage is the collection of all sieves generated by singletons $\{f\}\quad.$

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item The objects of the Schanuel topos are called [[nominal set|nominal sets]] and correspond precisely to the pullback preserving functors $FinSet_{mono}\to Set$.


\item The Schanuel topos $Sh(FinSet^{op}_{mono},J)$ is [[atomic topos|atomic]] over $Set$ ([[Stephen Schanuel|S. Schanuel]], cf. (Wraith 1978), p.335) hence Boolean. This fact can be viewed as a reflex of the \emph{urelements} in Fraenkel-Mostowski set theory.


\item $Sh(FinSet^{op}_{mono},J)$ is the [[classifying topos]] $Set[D_\infty]$ for the [[theory of infinite decidable objects]] $D_\infty$ i.e. for a Grothendieck topos $\mathcal{E}$ geometric morphisms $\mathcal{E}\to Sh(FinSet^{op}_{mono},J)$ correspond to infinite [[decidable objects]] in $\mathcal{E}$. $Sh(FinSet^{op}_{mono},J)$ is equivalent to $Sh_{\neg\neg}([FinSet_mono,Set])$.


\item $Sh(FinSet^{op}_{mono},J)$ is the category of continuous actions for the group of bijections of $N$ equipped with the topology derived from the product topology for $\prod_{N}N\quad .$


\item $Sh(FinSet^{op}_{mono},J)$ is the [[Kleisli category]] of the monad on the topos of [[species]] $Set^{FinSet_{iso}}$ induced by the inclusion of finite sets and bijections $FinSet_{iso}\hookrightarrow FinSet_{mono}$ (cf. Fiore-Menni 2004).



\end{itemize}
\hypertarget{remark}{}\subsubsection*{{Remark}}\label{remark}

For some information on the history of the Schanuel topos see section 10 of Menni (\hyperlink{Menni09}{2009, pp.529f}).

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

\begin{itemize}%
\item [[nominal set]]


\item [[locally decidable topos]]


\item [[theory of decidable objects]]


\item [[Fraenkel-Mostowski model]]


\item [[permutation model]]


\item [[atomic topos]]


\item [[species]]



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

\begin{itemize}%
\item [[Marcelo Fiore|M. Fiore]], [[Matias Menni|M. Menni]], \emph{Reflective Kleisli Subcategories of the Category of Eilenberg-Moore Algebras for Factorization Monads} , TAC \textbf{15} no. 2 (2004) pp.40-65. (\href{http://www.tac.mta.ca/tac/volumes/15/2/15-02.pdf}{pdf})


\item M. J. Gabbay, [[Andrew Pitts|A. M. Pitts]], \emph{A new approach to abstract syntax with variable binding}, Formal Aspects of Computing \textbf{13} (2002) pp.341-363. (\href{http://www.cl.cam.ac.uk/~amp12/papers/newaas/newaas-jv.pdf}{draft})


\item [[Peter Johnstone]], \emph{Sketches of an [[Elephant]] vols. I,II}, Oxford UP 2002. (pp.79f, 691, 925)


\item [[F. William Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs} , Cont. Math. \textbf{92} (1989). (pp.298f)


\item [[Saunders Mac Lane|S. Mac Lane]], [[Ieke Moerdijk|I. Moerdijk]], \emph{Sheaves in Geometry and Logic} , Springer Heidelberg 1994. (pp.155, 158)


\item [[Matías Menni]], \emph{About N-quantifiers} , Appl. Cat. Struc. \textbf{11} (2003) pp.421-445. (\href{https://sites.google.com/site/matiasmenni/newQuantifiers.pdf?attredirects=0}{preprint})


\item [[Matías Menni]], \emph{Algebraic categories whose projectives are explicitly free} , TAC \textbf{22} no.20 (2009) pp.509-541. (\href{http://www.tac.mta.ca/tac/volumes/22/20/22-20abs.html}{abstract})


\item Sam Staton, \emph{Name-passing process calculi: operational models and structural operational semantics}, Technical Report \textbf{688} CL University Cambridge 2007. (\href{https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-688.pdf}{pdf})


\item [[Gavin Wraith|G. Wraith]], \emph{Intuitionistic Algebra: Some Recent Developments of Topos Theory} , Proc. ICM Helsinki (1978) pp.331-337. (\href{http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0331.0338.ocr.pdf}{pdf})



\end{itemize}


\end{document}