关于树与圆盘的对偶性
On the duality between trees and disks
摘要：
André Joyal引入了一个组合范畴，它在弱ω-范畴的定义中发挥了作用。他把范畴Theta定义为对偶的Theta。在随后的文献中，提供了对Theta的更具体的描述。本文给出了对偶等价的另一种证明，并介绍了与圆盘或θ等价的各种范畴，每个范畴都提供了一个有用的观点。在这第二个版本的文件的内容已经重新组织的目标是一个更可读的介绍。我们定义了扩充范畴和它们的缩减对应物（缺少扩充范畴的单个平凡对象）。这些增广范畴更适合于归纳论证，它们的约化对应物等价于圆盘和θ。圆盘和θ之间的等价性在第4节和第6节中使用从区间和序数中归纳定义的范畴（在第3节中）进行了证明。最后两节采取了一种更范畴的观点，构造了所谓的标记树的范畴，并表明它们等价于它们的归纳定义的对应物，也等价于Disk和Theta。增加和减少类别之间的区别纠正了第一版中的一个错误，即终端树包含在磁盘类别中。
英文摘要：
A combinatorial category Disks was introduced by André Joyal to play a role in his definition of weak omega-category. He defined the category Theta to be dual to Disks. In the ensuing literature, a more concrete description of Theta was provided. In this paper we provide another proof of the dual equivalence and introduce various categories equivalent to Disk or Theta, each providing a helpful viewpoint. In this second version the paper's contents have been reorganized with the goal of a more readable presentation. We define augmented categories and their reduced counterparts (which lack a single trivial object of the augmented category). These augmented categories are more suitable for inductive arguments and their reduced counterparts are equivalent to Disk and Theta. The equivalence between Disk and Theta is demonstrated in Sections 4 and 6 using categories inductively defined (in Section 3) from intervals and ordinals. The last two sections take a more categorical perspective, constructing categories of so-called labeled trees and showing that they are equivalent to their inductively defined counterparts, and so to Disk and Theta. The distinction between augmented and reduced categories corrects an error in the first version where the terminal tree was included in the category Disk.
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