btw does anyone know where to find anything about higher categories with morphisms at transfinite ordinal levels? or is this not coherent or not explored because there's no motivation or something. i think it makes sense (if the underlying shapes use degeneracies and projections offsetting by some variable amount rather than always as compositions of those of one dimension, with the additional required identities that brings) but haven't been able to find anything in searching
i think that transfinite depths aren't necessarily incoherent, but definitely unmotivated. speaking as a nonexpert, it seems like there are two primary groups interested in infty-cats: the homotopy theorists, who bring compelling examples of weak infty-groupoids that arise "in nature" and are mostly concerned with grokking homotopy types; and the higher category people, who have an innate yearning for categorification and for whom infty-cats are a sort of corecursive object. so neither party really needs any more than infty-cats.