text: 401,98,1457669678596300803,958374804,Jonathan Gorard,"<LINK> New paper (with Xerxes) on the homotopic foundations of @wolframphysics , addressing the fundamental question of *how* and *why* geometrical structures in physics (such as spacetime, Hilbert space, etc.) emerge from discrete ""pregeometric"" data! (1/5) <LINK> The basic idea is that, starting from a multiway system, which is really a monoidal 1-category, one can perform “completions” by introducing higher cells (and thus higher homotopies) until one obtains the full ""rulial"" multiway system, which is really an infinity-groupoid. (2/5) The “rulial” multiway system can thus be interpreted as a homotopy type via Grothendieck's hypothesis, and the “multiverse” of all such “rulial” multiway systems thus carries the structure of a cohesive (infinity, 1)-topos, with cohesivity being preserved under fibration. (3/5) Treating “rulial space” as a fibration, one can then obtain the usual discrete structures (hypergraphs, causal graphs, multiway systems, etc.) as global sections of rulial space, which inherit spatial/geometrical structure functorially from the aforementioned homotopy type. (4/5) It's perhaps one of the most philosophically interesting papers I've ever (co-)written, since it begins to get at the foundational question of *why* the laws of physics have the structure that they do. With potential applications to TQFT, (higher) gauge field theory, etc. (5/5) @mattecapu @wolframphysics Cheers Matteo! Since ultimately we were concerned with obtaining homotopy n-types, it made sense to work with n-fold groupoids (since in that case it doesn’t matter). We could always collapse the faces in the cells to obtain the corresponding n-categories if we wanted to…",https://arxiv.org/abs/2111.03460,"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. ","Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy