从Wolfram模型重写系统为同伦型的前几何空间
Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types
摘要：
空间是如何从由计算规则支配的前几何离散积木中产生的？为了解决这个问题，我们研究Wolfram模型的非确定性重写系统（多路系统）。我们将这些重写系统表示为同伦类型。使用这个新的配方，我们概述了空间结构如何可以functionally继承前几何类型理论的建设。我们展示了如何从重写规则构造更高的同伦类型。这些对应于$n$ -fold范畴的态射。随后，Wolfram模型规则多路系统的$n \to \infty$极限被确定为一个$\infty$ -广群，后者与Grothendieck的同伦假设相关。然后，我们继续展示这个建设如何扩展到rulial多路系统的分类空间，它形成了一个多路系统的多重宇宙，并进行正式结构的${\left(\infty, 1\right)}$ -拓扑。这种与更高范畴结构的对应关系为理解与物理学相关的空间如何从前几何组合模型中产生提供了一种新的方法。一个关键的问题，我们已经解决了这里是抽象的非确定性重写系统更高的同伦空间。综合地构造空间和几何的结果是，它消除了关于模型的几何属性的特别假设，例如先验背景或预先分配的几何数据。相反，几何结构是由更高的结构功能继承的。这与量子引力模型所采用的基本时空离散化的不同选择的形式合理性有关。最后，我们的评论如何我们的框架高阶范畴理论的组合结构，证实与其他方法调查更高的范畴结构相关的物理基础。
英文摘要：
How do spaces emerge from pregeometric discrete building blocks governed by computational rules? To address this, we investigate non-deterministic rewriting systems (multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation, we outline how spatial structures can be functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopy types are constructed from rewriting rules. These correspond to morphisms of an $n$-fold category. Subsequently, the $n \to \infty$ limit of the Wolfram model rulial multiway system is identified as an $\infty$-groupoid, with the latter being relevant given Grothendieck's homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an ${\left(\infty, 1\right)}$-topos. This correspondence to higher categorical structures offers a new way to understand how spaces relevant to physics may arise from pregeometric combinatorial models. A key issue we have addressed here is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it eliminates ad hoc assumptions about geometric attributes of a model such as an a priori background or pre-assigned geometric data. Instead, geometry is inherited functorially by higher structures. This is relevant for formally justifying different choices of underlying spacetime discretization adopted by models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions, corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.
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