March 14, 2020

Answered by: Jonathan Gorard

How do your models relate to homotopy theory, derived geometry and higher category theory?

The relationship between the Wolfram model and ordinary category theory is actually relatively straightforward. One can think of a given hypergraph substitution system as being a morphism of the category Set, mapping the category of possible hypergraphs onto a power set construction on the category of possible hypergraphs, where the power set construction is considered to be an endofunctor on the category Set. As such, a Wolfram model system is really just an F-coalgebra of the power set functor.
The relationship to higher-order mathematics, specifically homotopy theory and higher category theory, and their geometrical incarnation in the form of derived geometry, is somewhat more speculative. A possible connection exists via the “snake states” of multiway evolution, as described in our answer to the question about string theory above. The essential idea here would be to use the so-called “cobordism hypothesis”—a theorem which implies that functors on monoidal (, n)-categories are entirely determined by their values on a single point, corresponding to an n-vector space of states—to deduce, for instance, that whilst the evolution of a single eigenstate through the multiway evolution graph may be described by a 1-category of spaces of states, the evolution of a maximally-consistent snake state may be described by a higher-order n-category of n-modules.