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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 comes with an adjunction where the left adjoint stems from profunctor composition and the right adjoint is defined for presheaves and as i.e. the set of natural transformations from to or, in other words, in .

   For an endoprofunctor define a \textsl{Jónsson-Tarski M-algebra} as a pair where and a natural isomorphism . The resulting category is a topos since a site can be constructed from by adjoining new arrows for each and with covers the set of these arrows (cf. Worrell 2002, Leinster 2007). Furthermore, is monadic over .

   The classical Jónsson-Tarski topos arises from this process by taking assigning to the unique object of the terminal category the two element set . A presheaf on is just a set whence is just the set of all maps i. e. . The site for it is the free category generated by the graph with one node and two edges i. e. the free monoid on two generators and with coverage .

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.