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    • CommentRowNumber1.
    • CommentAuthorThomas Holder
    • CommentTimeJul 7th 2014

    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.

    • CommentRowNumber2.
    • CommentAuthorspitters
    • CommentTimeJul 7th 2014

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

    • CommentRowNumber3.
    • CommentAuthorThomas Holder
    • CommentTimeJul 7th 2014

    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.

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeApr 4th 2022

    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:𝒜M:\mathcal{A} ⇸\mathcal{B} comes with an adjunction M[M,]:Set 𝒜 opSet op-\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 XSet opX\in Set^{\mathcal{B}^{op}} and a𝒜a\in \mathcal{A} as [M,X](a):=Nat(M(,a),X)[M,X](a):=Nat(M(-,a),X) i.e. the set of natural transformations from M(,a)M(-,a) to XX or, in other words, Hom(M(,a),X)Hom(M(-,a),X) in Set opSet^{\mathcal{B}^{op}}.

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

    The classical Jónsson-Tarski topos 𝒥 2\mathcal{J}_2 arises from this process by taking M=2:11M=\mathbf{2}:\mathbf{1} ⇸\mathbf{1} assigning to the unique object \bullet of the terminal category 1\mathbf{1} the two element set {,*}\{\star,\ast\}. A presheaf on 1\mathbf{1} is just a set XX whence [2,X]()[\mathbf{2}, X](\bullet) is just the set of all maps {,*}X\{\star,\ast\}\to X i. e. X×XX\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\}.

    • CommentRowNumber5.
    • CommentAuthorThomas Holder
    • CommentTimeApr 27th 2022
    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.