Gálvez-Kock-Tonks 猜想在局部离散分解空间上的情形
The Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces
AI总结：
本文证明了 Gálvez-Kock-Tonks 猜想在局部离散分解空间上的情形，通过构造局部严格模型并利用 2-范畴论证，表明 Lawvere 区间构造仅允许恒等自修改，为一般猜想提供了首个实质证据。
摘要：
Gálvez-Carrillo、Kock 和 Tonks 构造了一个由所有 Möbius 区间组成的分解空间 $U$，作为 Lawvere 区间构造在 Möbius 范畴中的接收者，并猜想 $U$ 具有某种普适性质：对于每个 Möbius 分解空间 $X$，从 $X$ 到 $U$ 的 culf 函子空间是可缩的。在本文中，我们在同伦 1-型的层面上工作，证明了该猜想的第一个情形，即局部离散分解空间的情形。这也为一般猜想提供了第一个实质性的证据。该情形足够广泛，可以覆盖所有局部有限偏序集、Cartier--Foata 幺半群、Möbius 范畴以及严格（有向）限制物种。证明是 2-范畴的。首先，我们构造了 $U$ 的一个局部严格模型，然后用它来手工证明，Lawvere 区间构造作为自然变换，不允许除恒等变换之外的其他自修改。
英文摘要：
Gálvez-Carrillo, Kock, and Tonks constructed a decomposition space $U$ of all Möbius intervals, as a recipient of Lawvere's interval construction for Möbius categories, and conjectured that $U$ enjoys a certain universal property: for every Möbius decomposition space $X$, the space of culf functors from $X$ to $U$ is contractible. In this paper, we work at the level of homotopy 1-types to prove the first case of the conjecture, namely for locally discrete decomposition spaces. This provides also the first substantial evidence for the general conjecture. This case is general enough to cover all locally finite posets, Cartier--Foata monoids, Möbius categories and strict (directed) restriction species. The proof is 2-categorical. First, we construct a local strict model of $U$, which is then used to show by hand that the Lawvere interval construction, considered as a natural transformation, does not admit other self-modifications than the identity.
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。
