We study when co-evolving (or adaptive) higher-order networks defined on directed hypergraphsadmit a simplicial description. Binary and triadic couplings are modelled by time-dependent weighttensors. Using representation theory of the symmetric group Sk, we decompose these tensors into fullysymmetric, fully antisymmetric, and mixed isotypic components, and track their Frobenius norms todefine three asymptotic regimes and a quantitative notion of convergence. In the symmetric (resp.antisymmetric) limit, we certify emergence and stability of simplicial complexes via a local boundarytest and interior drift conditions that enforce downward-closure; in the mixed limit, we show that theminimal faithful object is a semi-simplicial set. We illustrate the theory with simulations that trackthe isotypic Frobenius norms and the higher-order structure. Practically, our work provides rigorousconditions under which homological tools are justified for adaptive higher-order systems.