text:   Types",6,"['<LINK>\nNew 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…']",21,11,1684