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\begin{document}

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\section*{Jónsson-Tarski topos}

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

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

[[!include topos theory - contents]]

\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{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{rosenthals_approach}{Rosenthal's approach}\dotfill \pageref*{rosenthals_approach} \linebreak
\noindent\hyperlink{jnssontarski_toposes_and_selfsimilarity}{J\'o{}nsson-Tarski toposes and self-similarity}\dotfill \pageref*{jnssontarski_toposes_and_selfsimilarity} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{J\'o{}nsson-Tarski topos} is `the topos analogue of the [[Cantor space]]'.\footnote{Quote from Bunge\&Funk (2006, p.183).} 

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

The \textbf{J\'o{}nsson-Tarski topos} $\mathcal{J}_2$ is the category of [[Jónsson-Tarski algebras]] considered as topos, i.e. its objects are sets $X$ together with an isomorphism $X\to X\times X$ and morphisms are functions that commute with the structure isomorphisms.

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

\begin{itemize}%
\item $\mathcal{J}_2$ is an example of an [[algebraic variety]] that is also a topos (cf. Johnstone 1985).


\item $\mathcal{J}_2=Sh(M_2,J)$ where $M_2$ is the free monoid on two generators $a, b$ and $J$ is the [[coverage]] whose only covering family on the unique object $\cdot$ of $M_2$ is $\{a:\cdot\rightarrow \cdot ,b:\cdot\rightarrow\cdot\}$. So $\mathcal{J}_2$ is in a fact a [[Grothendieck topos]].


\item $M_2$, as a free monoid, is cancellative, hence all-monic and, accordingly, $\mathcal{J}_2$ is an [[étendue]] ([[Peter Freyd]]). It is discussed from this [[étendue|perspective]] as a [[petit topos]] for \emph{labeled graphs} in (Lawvere 1989).


\item Actually, Freyd observed that $\mathcal{J}_2/F(1)\cong Sh(2^\mathbb{N})$ with $F(1)$ the free J\'o{}nsson-Tarski algebra on one generator and $2^\mathbb{N}$ the [[Cantor space]] - this motivates the above quote from Bunge\&Funk: $\mathcal{J}_2$ looks locally like (the sheaf topos on) $2^\mathbb{N}$ ! $Sh(2^\mathbb{N})$ classifies ` subsets of $\mathbb{N}$ ` in the sense that for cocomplete $\mathcal{E}$ geometric morphisms $\mathcal{E}\to Sh(2^\mathbb{N})$ correspond to morphisms $\Delta(\mathbb{N})\to\Delta(2)$ in $\mathcal{E}$, with $\Delta$ the [[constant sheaf functor]] (cf. \hyperlink{MacLaneMoerdijk}{Mac Lane\& Moerdijk}, ex.VIII.10, pp.470-71).



\end{itemize}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

The idea to consider generalizations of $\mathcal{J}_2$ seemed to have appeared first in the context of work on \'e{}tendues (Rosenthal 1981, Lawvere 1989).

\hypertarget{rosenthals_approach}{}\subsubsection*{{Rosenthal's approach}}\label{rosenthals_approach}

K. Rosenthal (1981) starts from two basic facts about \'e{}tendues, namely that $Set^{\mathcal{C}^{op}}$ is an [[étendue]] iff all morphisms in $\mathcal{C}$ are monic, and that $Sh(\mathcal{E},J)$ is an \'e{}tendue if $\mathcal{E}$ is an \'e{}tendue. His goal is to construct \'e{}tendues from an all-monic $\mathcal{C}$ by defining a topology $J$ on $Set^{\mathcal{C}^{op}}$ from a functor $H:\mathcal{C}\to Set$ satisfying:

\begin{enumerate}%
\item $H(f)$ is monic ,and


\item if $x\in Im(H(f))\cap Im(H(g))$ then there is $k\in\mathcal{C}$ with $k\leq f$ and $k\leq g$ such that $x\in Im(H(k))$.



\end{enumerate}
Now given $X\in\mathcal{C}$ and a sieve $B\in\Omega(X)$ define a sieve

$j_X(B):=\{\quad f\in\mathcal{C}| cod(f)=X\quad\wedge\quad Im(H(f))\subseteq\bigcup _{g\in B} Im(H(g))\quad\}.$

The resulting map $j:\Omega\to\Omega$ is a topology.

For $\mathcal{C}^{op}=\mathcal{M}_2$, the free monoid on two generators, and $H(X)=2^\mathbb{N}$, the functor constantly the Cantor set, this yields $\mathcal{J}_2$.

The generalization to $\mathcal{C}^{op}=\mathcal{M}_\infty$, the free monoid on countably infinite many generators, and the [[Baire space of sequences|Baire space]] $\mathbb{B}=\mathbb{N}^\mathbb{N}$ exhibits the \textbf{infinite J\'o{}nsson-Tarski topos} $\mathcal{J}_\infty$, i.e. the category of sets $A$ with an isomorphism to $A^\mathbb{N}$, as $Sh(\mathbb{N}^\mathbb{N})$ locally.

\hypertarget{jnssontarski_toposes_and_selfsimilarity}{}\subsubsection*{{J\'o{}nsson-Tarski toposes and self-similarity}}\label{jnssontarski_toposes_and_selfsimilarity}

Work on a categorical concept of self-similarity led [[Tom Leinster|T. Leinster]] (2007) to another generalization of the J\'o{}nsson-Tarski topos.

The first hint to a connection stems from the self-similarity system $M:\mathbf{1}&#8696;  \mathbf{1}$ with $M=\{0,1\}$ which is just a profunctorial instruction to paste two copies of a space $X$ together and the universal solution none other than the [[Cantor space]] $2^\mathbb{N}$.

Caveat: this entry is still under construction!

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Jónsson-Tarski algebra]]


\item [[Cantor space]]



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

\begin{itemize}%
\item [[Marta Bunge]], [[Jonathon Funk]], \emph{Singular Coverings of Toposes} , Springer LNM vol. 1890, Heidelberg 2006.


\item [[Peter Johnstone]], \emph{When is a Variety a Topos?} , Algebra Universalis \textbf{21} (1985) pp.198-212.


\item [[Peter Johnstone]], \emph{Collapsed Toposes as Bitopological Spaces} , pp.19-35 in \emph{Categorical Topology} , World Scientific Singapore 1989.


\item [[Peter Johnstone]], \emph{Collapsed Toposes and Cartesian Closed Varieties} , JA \textbf{129} (1990) pp.446-480.


\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] vol. I} , Oxford UP 2002. (sec. A2.1, p.80)


\item [[Peter Johnstone|P. T. Johnstone]], A. J. Power, T. Tsujishita, H. Watanabe, J. Worrell, \emph{The structure of categories of coalgebras} , Theoret. Comp. Sci \textbf{260} (2001) pp.87-117.


\item [[F. William Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs} , Cont. Math. \textbf{92} (1989) pp.261-299.


\item [[Tom Leinster]], \emph{J\'o{}nsson-Tarski toposes}, Talk Nice 2007. (\href{http://www.maths.ed.ac.uk/~tl/nice/jt.pdf}{slides})


\item [[Saunders Mac Lane|S. Mac Lane]], [[Ieke Moerdijk|I. Moerdijk]], \emph{Sheaves in Geometry and Logic} , Springer Heidelberg 1994. (pp.470-471)


\item [[Kimmo I. Rosenthal]], \emph{\'E{}tendues and Categories with Monic Maps} , JPAA \textbf{22} (1981) pp.193-212.


\item James Worrell, \emph{A Note on Coalgebras and Presheaves} , Electronic Notes in Theoretical Computer Science \textbf{65} no.3 (2003) pp.1-10.



\end{itemize}
[[!redirects Jonsson-Tarski topos]] [[!redirects Jonsson-Tarski Topos]]



\end{document}