Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories (2105.10822v2)

Abstract: We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order $n$ may naturally be formalized as an $n$-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ${\infty}$-groupoid. Via Grothendieck's homotopy hypothesis, this ${\infty}$-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of homotopical multiway systems may potentially be used for making formal connections to homotopy spaces upon which models relevant to physics may be instantiated.

• The paper demonstrates that multiway rewriting systems can be organized as n-fold categories by leveraging higher homotopies.
• It introduces a computational framework that inserts bi-directional homotopies between paths, ensuring a robust combinatorial structure.
• The work bridges discrete rewriting systems with topological and categorical concepts, paving the way for applications in physics and advanced mathematics.

Homotopies in Multiway Rewriting Systems: An Insight into $n$-Fold Categories

The paper "Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories" explores the algebraic and compositional characteristics of abstract multiway rewriting systems within the Wolfram model framework. It explores the emergence of higher homotopies in these systems, establishing that such structures can be formalized as $n$-fold categories and ultimately lead to an ${\infty}$-groupoid when equipped with invertible rewriting relations. This abstract formalism aims to link these systems to the conceptual underpinnings of homotopy spaces, suggesting potential applications in formalizing physical models.

Core Contributions and Findings

The paper provides a comprehensive examination of how homotopical maps arise within multiway rewriting systems through the inclusion of higher-order rewriting rules sourced from an abstract rulial space. Notably, the authors put forth the proposition that multiway systems with homotopies up to order $n$ may be systematically organized as $n$-fold categories. This construction facilitates the corresponding $(n-1)$-fold groupoid structure when inverse morphisms are incorporated.

Key sections of the paper offer a rigorous treatment of multiway systems, underscoring their role as non-deterministic abstract rewriting entities with a partial causal order. Such systems are described compositionally via $F$-coalgebras, analogous to categories of tuples represented in terms of power sets.

The article details the process of constructing these homotopical structures using a computational framework. A central algorithmic procedure is delineated, whereby bi-directional edges (interpreted as homotopies) are introduced between paths connecting the same vertices within the multiway evolution graph, symbolizing multiple proofs of a proposition with similar endpoints.

Theoretical Implications

From a theoretical standpoint, the findings bolster the mathematical foundation needed for connecting discrete computational systems with higher-dimensional category theory. The conversion of multiway rewriting systems into $n$-fold categories highlights the potential of these systems to be foundational constructs for significant mathematical objects like topological spaces and homotopy theories.

The transformation of these rewriting systems into ${\infty}$-groupoids, within the confines of Grothendieck's homotopy hypothesis, underscores profound implications for understanding space as a composite of purely combinatorial elements. This suggests that discrete rewriting systems can model complex systems within physics using synthetic geometric principles.

Practical Implications and Future Directions

This paper lays the groundwork for incorporating computational and abstract algebraic structures into the exploration of synthetic geometry and quantum field theories. The framework could potentially lead to new avenues in understanding the theoretical structure of cohesive $\infty$-topoi.

In future research, extending these computations to incorporate additional real-world constraints and models may help flesh out the Wolfram model's ambitions to act as a unified framework for physics. Addressing the computability and completeness of higher-categorical constructs within these rewriting systems will also remain a relevant pursuit.

Overall, the methodology and insights presented in this paper reinforce the depth at which algebraic and computational narratives intertwine, particularly within the realms of topology and physics. This paper, by formalizing the computational underpinnings of homotopical multiway systems, sets a foundation for further computational applications and theoretical explorations within modern mathematics and theoretical physics.