儿童平面Heyting代数2：局部算子，J-算子和Slashing
Planar Heyting Algebras for Children 2: Local Operators, J-Operators, and Slashings
摘要：
选择一个拓扑$E$。在$E$上有几种不同的“层性概念”。我们如何将它们形象化？让我们把$E$的分类器对象称为$Ω$，把它的真值海廷代数$Sub(1_E)$称为$H$ ;我们有时会把$H$称为拓扑的“逻辑”。有一种众所周知的方法可以将层性的概念表示为态射$j:Ω\to Ω$，但是当我们显式地绘制这些“$j$”时，它们会产生大图;在这里，我们将看到一种更易于管理的方法，可以将这些“$j$”表示为映射$J:H\to H$。在本系列的前一篇文章中，我们展示了如何使用海廷代数的某些玩具模型（称为“ZHA”）来开发海廷代数和直觉主义命题逻辑如何工作的视觉直觉;在这里，我们将扩展到层。完整的想法是这样的：层性的概念对应于局部算子，反之亦然;局部算子对应于J-算子，反之亦然;如果我们的海廷代数$H$是ZHA，则J-算子对应于$H$上的斜线，反之亦然; $H$上的斜线对应于“问号集”，反之亦然，并且每组问号引出擦除和重构的概念，这引出层。而且，每个ZHA $H$对应于（非循环的）2列图，反之亦然，并且对于任何2列图$(P,A)$，拓扑$\mathbf{Set}^{(P,A)}$的逻辑恰好是与$(P,A)$相关联的ZHA $H$。
英文摘要：
Choose a topos $E$. There are several different "notions of sheafness" on $E$. How do we visualize them? Let's refer to the classifier object of $E$ as $Ω$, and to its Heyting Algebra of truth-values, $Sub(1_E)$, as $H$; we will sometimes call $H$ the "logic" of the topos. There is a well-known way of representing notions of sheafness as morphisms $j:Ω\to Ω$, but these `$j$'s yield big diagrams when we draw them explicitly; here we will see a way to represent these `$j$'s as maps $J:H\to H$ in a way that is much more manageable. In the previous paper of this series we showed how certain toy models of Heyting Algebras, called "ZHAs", can be used to develop visual intuition for how Heyting Algebras and Intuitionistic Propositional Logic work; here we will extend that to sheaves. The full idea is this: notions of sheafness correspond to local operators and vice-versa; local operators correspond to J-operators and vice-versa; if our Heyting Algebra $H$ is a ZHA then J-operators correspond to slashings on $H$, and vice-versa; slashings on $H$ correspond to "sets of question marks" and vice-versa, and each set of question marks induces a notion of erasing and reconstructing, which induces a sheaf. Also, every ZHA $H$ corresponds to an (acyclic) 2-column graph, and vice-versa, and for any two-column graph $(P,A)$ the logic of the topos $\mathbf{Set}^{(P,A)}$ is exactly the ZHA $H$ associated to $(P,A)$.
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