每个闭包运算符都导出一个拓扑，反之亦然（“子版本”）
Each closure operator induces a topology and vice-versa ("version for children")
摘要：
理解基本拓扑上的层的一个主要前提是证明拓扑上的（Lawvere-Tierney）拓扑诱导其上的闭包算子，反之亦然。这个标准定理通常以相对简短的方式呈现，大多数细节留给读者，并且没有提示如何可视化一些最难的公理和证明。在第一个层次上，这些注释试图以所有细节和视觉方式呈现标准定理，遵循“关于我最喜欢的在范畴论中绘制缺失图的约定“[Ochs 2020];特别是，一些性质，如拉回稳定性，总是以相同的“形状”绘制。在第二个层面上，这些笔记也是一个实验，在“原型案例”上做这些证明，使所有的证明容易提升到“一般案例”。我们的第一个原型案例是“包含拓扑结构”。这是来自[Lambek/Scott 1986]的“具有规范子范畴的拓扑”的变体;所有形式为$\mathbf{Set}^\mathbf{C}$的拓扑，其中$\mathbf{C}$是一个小范畴，都是具有包含的拓扑，当我们处理具有包含的拓扑时，像子集和交集这样的概念很容易形式化。我们做我们所有的证明之间的对应关系封闭运营商和拓扑toposes与包含，然后我们展示了如何提升我们所有的证明，证明工作在任何topos。我们的第二个原型情况是$\mathbf{Set}^\mathbf{D}$形式的拓扑，其中$\mathbf{D}$是一个有限的两列图。我们展示了一种方法来可视化所有的LT-拓扑的形式$\mathbf{Set}^\mathbf{D}$的toposes，我们正式定义了一种方法“添加视觉直观的证明有关toposes”，这是有关的几个技术做“范畴论的儿童”中解释的“在我最喜欢的公约. ".
英文摘要：
One of the main prerequisites for understanding sheaves on elementary toposes is the proof that a (Lawvere-Tierney) topology on a topos induces a closure operator on it, and vice-versa. That standard theorem is usually presented in a relatively brief way, with most details being left to the reader and with no hints on how to visualize some of the hardest axioms and proofs. These notes are, on a first level, an attempt to present that standard theorem in all details and in a visual way, following the conventions in "On my favorite conventions for drawing the missing diagrams in Category Theory" [Ochs2020]; in particular, some properties, like stability by pullbacks, are always drawn in the same "shape". On a second level these notes are also an experiment on doing these proofs on "archetypal cases" in ways that makes all the proofs easy to lift to the "general case". Our first archetypal case is a "topos with inclusions". This is a variant of the "toposes with canonical subobjects" from [Lambek/Scott 1986]; all toposes of the form $\mathbf{Set}^\mathbf{C}$, where $\mathbf{C}$ is a small category, are toposes with inclusions, and when we work with toposes with inclusions concepts like subsets and intersections are very easy to formalize. We do all our proofs on the correspondence between closure operators and topologies in toposes with inclusions, and then we show how to lift all our proofs to proofs that work on any topos. Our second archetypal case is toposes of the form $\mathbf{Set}^\mathbf{D}$, where $\mathbf{D}$ is a finite two-column graph. We show a way to visualize all the LT-topologies on toposes of the form $\mathbf{Set}^\mathbf{D}$, and we define formally a way to "add visual intuition to a proof about toposes"; this is related to the several techniques for doing "Category Theory for children" that are explained in "On my favorite conventions...".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
