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1. • CommentRowNumber1.
   • CommentAuthorThomas Holder
   • CommentTimeJul 7th 2014
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   Author: Thomas Holder
   Format: MarkdownItexI have created a separate entry for [[Jonsson-Tarski topos]] for which I've deconstructed [[Jonsson-Tarski algebra]]. Hope this is ok with the previous author. As I brood over JT right now, I'll be collecting further information in the entry in the coming days. I have also created a separate entry for [[etendue]] but this is a bit of a mess for the time being and would necessitate to reflect the thread on etendu/e we had here in spring. for the moment, I can't give much attention to 'etendue' though. At one moment this should be harmonized with [[localic topos]].

   I have created a separate entry for Jonsson-Tarski topos for which I’ve deconstructed Jonsson-Tarski algebra. Hope this is ok with the previous author. As I brood over JT right now, I’ll be collecting further information in the entry in the coming days. I have also created a separate entry for etendue but this is a bit of a mess for the time being and would necessitate to reflect the thread on etendu/e we had here in spring. for the moment, I can’t give much attention to ’etendue’ though. At one moment this should be harmonized with localic topos.

2. • CommentRowNumber2.
   • CommentAuthorspitters
   • CommentTimeJul 7th 2014
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   Author: spitters
   Format: MarkdownItexTom Leinster discusses JT-toposes (plural). Maybe your text can reflect this?

   Tom Leinster discusses JT-toposes (plural). Maybe your text can reflect this?

3. • CommentRowNumber3.
   • CommentAuthorThomas Holder
   • CommentTimeJul 7th 2014
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   Author: Thomas Holder
   Format: MarkdownItexIt will! my symbol for the JT topos is already equipped with a counting index. Already at this low number a strange beast though! what particularly intrigues me, is the question whether there is a relation between the internal language and MSO.

   It will! my symbol for the JT topos is already equipped with a counting index. Already at this low number a strange beast though! what particularly intrigues me, is the question whether there is a relation between the internal language and MSO.

4. • CommentRowNumber4.
   • CommentAuthorGuest
   • CommentTimeApr 4th 2022
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   Author: Guest
   Format: MarkdownItexComing back to this "construction site entry" it is time to fill in a proper account of Tom Leinster's approach and the following should eventually replace the section on "Jónsson-Tarski toposes and self-similarities": ####Jónsson-Tarski toposes via profunctors#### Work on a categorical concept of self-similarity led T. Leinster (2007) to another generalization of the Jónsson-Tarski topos. He starts from the observation that any [[profunctor]] $M:\mathcal{A} ⇸\mathcal{B}$ comes with an [[adjunction]] $-\otimes M\dashv [ M,-]:Set^{\mathcal{A}^{op}}\to Set^{\mathcal{B}^{op}}$ where the left adjoint stems from profunctor composition and the right adjoint is defined for presheaves $X\in Set^{\mathcal{B}^{op}}$ and $a\in \mathcal{A}$ as $[M,X](a):=Nat(M(-,a),X)$ i.e. the set of [[natural transformation|natural transformations]] from $M(-,a)$ to $X$ or, in other words, $Hom(M(-,a),X)$ in $Set^{\mathcal{B}^{op}}$. For an endoprofunctor $M:\mathcal{A} ⇸\mathcal{A}$ define a \textsl{Jónsson-Tarski M-algebra} as a pair $(X,\xi)$ where $X\in Set^{\mathcal{A}^{op}}$ and $\xi$ a [[natural isomorphism]] $X\overset{\simeq}{\to} [M,X]$. The resulting category $\mathcal{J}_M$ is a topos since a site $(\mathcal{A}_M,J)$ can be constructed from $\mathcal{A}$ by adjoining new arrows $b\overset{m}{\to} a$ for each $b,a\in\mathcal{A}$ and $m\in M(b,a)$ with covers $J(a)$ the set of these arrows (cf. Worrell 2002, Leinster 2007). Furthermore, $\mathcal{J}_M$ is monadic over $Set^{\mathcal{A}^{op}}$. The classical Jónsson-Tarski topos $\mathcal{J}_2$ arises from this process by taking $M=\mathbf{2}:\mathbf{1} ⇸\mathbf{1}$ assigning to the unique object $\bullet$ of the terminal category $\mathbf{1}$ the two element set $\{\star,\ast\}$. A presheaf on $\mathbf{1}$ is just a set $X$ whence $[\mathbf{2}, X](\bullet)$ is just the set of all maps $\{\star,\ast\}\to X$ i. e. $X\times X$. The site for it is the free category generated by the graph with one node $\bullet$ and two edges $\star,\ast$ i. e. the free monoid on two generators and with coverage $\{\star,\ast\}$.

   Coming back to this “construction site entry” it is time to fill in a proper account of Tom Leinster’s approach and the following should eventually replace the section on “Jónsson-Tarski toposes and self-similarities”:

   Jónsson-Tarski toposes via profunctors

   Work on a categorical concept of self-similarity led T. Leinster (2007) to another generalization of the Jónsson-Tarski topos. He starts from the observation that any profunctor $M:\mathcal{A} ⇸\mathcal{B}$ comes with an adjunction $-\otimes M\dashv [ M,-]:Set^{\mathcal{A}^{op}}\to Set^{\mathcal{B}^{op}}$ where the left adjoint stems from profunctor composition and the right adjoint is defined for presheaves $X\in Set^{\mathcal{B}^{op}}$ and $a\in \mathcal{A}$ as $[M,X](a):=Nat(M(-,a),X)$ i.e. the set of natural transformations from $M(-,a)$ to $X$ or, in other words, $Hom(M(-,a),X)$ in $Set^{\mathcal{B}^{op}}$.

   For an endoprofunctor $M:\mathcal{A} ⇸\mathcal{A}$ define a \textsl{Jónsson-Tarski M-algebra} as a pair $(X,\xi)$ where $X\in Set^{\mathcal{A}^{op}}$ and $\xi$ a natural isomorphism $X\overset{\simeq}{\to} [M,X]$. The resulting category $\mathcal{J}_M$ is a topos since a site $(\mathcal{A}_M,J)$ can be constructed from $\mathcal{A}$ by adjoining new arrows $b\overset{m}{\to} a$ for each $b,a\in\mathcal{A}$ and $m\in M(b,a)$ with covers $J(a)$ the set of these arrows (cf. Worrell 2002, Leinster 2007). Furthermore, $\mathcal{J}_M$ is monadic over $Set^{\mathcal{A}^{op}}$.

   The classical Jónsson-Tarski topos $\mathcal{J}_2$ arises from this process by taking $M=\mathbf{2}:\mathbf{1} ⇸\mathbf{1}$ assigning to the unique object $\bullet$ of the terminal category $\mathbf{1}$ the two element set $\{\star,\ast\}$. A presheaf on $\mathbf{1}$ is just a set $X$ whence $[\mathbf{2}, X](\bullet)$ is just the set of all maps $\{\star,\ast\}\to X$ i. e. $X\times X$. The site for it is the free category generated by the graph with one node $\bullet$ and two edges $\star,\ast$ i. e. the free monoid on two generators and with coverage $\{\star,\ast\}$.

5. • CommentRowNumber5.
   • CommentAuthorThomas Holder
   • CommentTimeApr 27th 2022
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   Author: Thomas Holder
   Format: TextThe following references should eventually added to [[Jónsson-Tarski algebra]] https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-62/issue-6/The-summer-meeting-in-Seattle/bams/1183521094.full https://math.stackexchange.com/questions/3544667/do-j%C3%B3nsson-tarski-algebras-form-a-schreier-variety and Simon Henry's remark https://mathoverflow.net/questions/396041/images-of-complemented-subobjects-in-hyperconnected-toposes-over-boolean-bases/396043?r=SearchResults&s=1|33.5942#396043 here.

   The following references should eventually added to Jónsson-Tarski algebra
   
   https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-62/issue-6/The-summer-meeting-in-Seattle/bams/1183521094.full
   
   https://math.stackexchange.com/questions/3544667/do-j%C3%B3nsson-tarski-algebras-form-a-schreier-variety
   
   and Simon Henry's remark
   
   https://mathoverflow.net/questions/396041/images-of-complemented-subobjects-in-hyperconnected-toposes-over-boolean-bases/396043?r=SearchResults&s=1|33.5942#396043
   
   here.