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I'm reading up on infinite generalizations of the fundamental theorem of distributive lattices. Wikipedia (June 15, 2017) says that there is a duality

between distributive lattices and coherent spaces (sometimes called spectral spaces)

where I have reproduced the links as they appear. I think that the article on coherent spaces they have linked to is referring to a different, unrelated notion. Could someone in logic check me on this before I correct it?

Thanks, and apologies if this is too frivolous a use of MO.

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  • $\begingroup$ I'm not a logician but it seems wrong to me. Coherent space, as defined in Johnstone's book, means a topological space where the quasi-compact open sets form a distributive lattice and a basis for the topology. $\endgroup$ – Benjamin Steinberg Jun 15 '17 at 15:25
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    $\begingroup$ Doesn't seem frivolous to me. $\endgroup$ – Nick Gill Jun 15 '17 at 15:25
  • $\begingroup$ Aren't distributive lattices the Ind-completion of finite distributive lattices which are dual to finite posets = finite $T_0$ spaces (ncatlab.org/nlab/show/distributive+lattice#OppositeCategory)? Making the dual of distributive lattices the Pro-completion of finite $T_0$-spaces which is one description given at spectral space? $\endgroup$ – Todd Trimble Jun 15 '17 at 16:17
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Yes, the notion of "coherent space" at the page you linked is completely unrelated to spectral spaces. The coherent spaces of that article were introduced by Jean-Yves Girard as a denotational semantics of second-order intuitionistic logic and were instrumental in his discovery of linear logic.

People have been aware of this potential confusion since the beginning of linear logic (see footnote 3, p.55 of Girard, Lafont and Taylor's book Proofs and Types), that is why some prefer to call them coherence spaces. I guess that the authors of the Wikipedia page didn't follow this practice.

Edit: Just after posting my answer, I realized that this potential confusion is explicitly pointed out in the Wikipedia page, as well as the alternative terminology in Proofs and Types.

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    $\begingroup$ Correct. I think the modified name coherence space was mine. I had previously learned about coherent spaces in the sense of Johnstone's book when I attended his Part III course on the material in his book. $\endgroup$ – Paul Taylor Jul 8 '17 at 21:30

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