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We investigate algebraic and compositional properties of abstract multiwayrewriting systems, which are archetypical structures underlying the formalismof the Wolfram model. We demonstrate the existence of higher homotopies in thisclass of rewriting systems, where homotopical maps are induced by the inclusionof appropriate rewriting rules taken from an abstract rulial space of allpossible such rules. Furthermore, we show that a multiway rewriting system withhomotopies up to order $n$ may naturally be formalized as an $n$-fold category,such that (upon inclusion of appropriate inverse morphisms via invertiblerewriting 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. Weconclude with some comments on how this computational framework of homotopicalmultiway systems may potentially be used for making formal connections tohomotopy spaces upon which models relevant to physics may be instantiated.

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