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 formalize 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 result from pregeometric combinatorial models. The key issue we have addressed here is to formally relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it removes 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 from globular structures. This is relevant for formally justifying different choices of underlying spacetime discretization adopted by various models of quantum gravity. Finally, we end with comments on how the framework of higher category-theoretic combinatorial constructions developed here, corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.
翻译:为了解决这个问题,我们调查沃尔夫拉姆模型的非确定性重写系统(多路系统),我们将这些重写系统正规化为同质类型。使用这一新公式,我们概述了空间结构如何从远地分解类型理论构造中传承。我们展示了如何从重写规则中构建更高的同质性类型。这些类型对应了美元-双倍类的物理结构形态。随后,我们调查沃尔夫拉姆模型多路系统的非确定性重写系统(多路系统)的美元-直径直径结构。随后,我们调查非确定性重写系统的美元-美元-直径模型(多路规则)的非确定性重写系统(多路系统)的变异性结构。关于美元-直径直径的直径直径直径直径结构(多路系统)的正态选择。这个更高级的直径直径直径系统(Wolfram)的背景被确定为美元-直径多路径系统结构的美元-直径直径直径直径直径系统,这一直径直径直径直径直径直径直径直径直径直径直径直径直径系统结构结构结构结构结构结构结构结构结构结构结构结构的直径直径分析模型的直径直径直径直径直径直径直径直对等模型的直对等模型的直径直径直径直向, 直向,从新的直到直径直径直向后向后向,直至右,从一个直至直路向,直径结构结构结构结构结构结构结构结构路向,直径直径直径直向后,从一个新的直径结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构结构