Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

Arsiwalla, Xerxes D.; Gorard, Jonathan; Elshatlawy, Hatem

doi:10.48550/arxiv.2105.10822

Cited by 1 publication

(4 citation statements)

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“…Subsequently, we prove that a multiway system equipped with homotopies up to order n may be formalized as an n-fold category, such that the infinite limit of this higherorder multiway system yields an ∞-groupoid (upon the admission of invertible rewriting rules). This section extends beyond initial work reported in [9], especially parts concerning (∞, 1)categories.…”

Section: Higher Homotopies In Multiway Rewriting Systems As N-fold Ca...mentioning

confidence: 74%

“…This article is an attempt at addressing two key objectives: Firstly, we seek to provide a mathematical framework for generating formal constructions of pregeometric structures, which, in certain to-be-specified limits, would yield candidate spaces within which topological or geometric structures relevant to physics may be modeled. The other objective of this work is to provide a new formal foundation for the Wolfram model [94], [93] (see also [48], [47], [49], [9]) using homotopy types. Both these issues are, of course, related.…”

Section: Introductionmentioning

confidence: 99%

“…These homotopies are induced by inclusion of 'higher-order' rules taken from a so-called 'rulial space', which designates the space of all possible rewriting rules of a given signature (for the case of graphs or hypergraphs, this space would comprise a monoidal category of cospans). Although a generic rewriting system may be thought of as simply being an F -coalgebra for the power set functor P (which, in the case of hypergraph rewriting, is further equipped with a natural symmetric monoidal structure), in earlier work [9], we showed that a rulial multiway rewriting system equipped with homotopies up to order n can elegantly be formalized as an n-fold category. Furthermore, upon including inverse morphisms (via the inclusion of invertible rewriting relations), the n → ∞ limit of the rulial multiway system yields an ∞-groupoid, which inherits the structure of a formal homotopy space via Grothendieck's homotopy hypothesis.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, we 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 (∞, 1)-topos. This correspondence to spaces and higher structures offers a formal understanding of how spatial structures relevant to theories of physics can be shown to originate from abstract combinatorial constructions [6], [9], [86], [16].…”

Section: Introductionmentioning

confidence: 99%

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Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

Arsiwalla¹,

Gorard²

2021

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How do spaces in physics emerge from pregeometric discrete building blocks governed by computational rules? To address this question we investigate non-deterministic rewriting systems (so-called multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation of the model, we motivate how spatial structures are functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopies are constructed from rewriting rules. These correspond to morphisms of an n-fold category. Subsequently, the n → ∞ limit of the Wolfram model rulial multiway system is identified as an ∞-groupoid, with the latter being relevant in light of 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 (∞, 1)-topos. This correspondence to higher categorical structures potentially offers a new way to understand how the kinds of spaces relevant to physics result from pregeometric combinatorial models. The key issue we have addressed in this work is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it removes the need to make ad hoc assumptions about geometric attributes of a model such as an a priori background or any pre-assigned geometric data. Instead, geometry is inherited functorially from higher structures. This can be particularly useful for formally justifying different choices of underlying spacetime discretization schemes adopted by various models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions closely corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.

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Section: Higher Homotopies In Multiway Rewriting Systems As N-fold Ca...mentioning

confidence: 74%