Topological statistical theory provides the foundation for a modern mathematical reformulation of classical statistical theory: Structural Statistics emphasizes the structural assumptions that accompany distribution families and the set of structure preserving transformations between them, given by their statistical morphisms. The resulting language is designed to integrate complicated structured model spaces like deep-learning models and to close the gap to topology and differential geometry. To preserve the compatibility to classical statistics the language comprises corresponding concepts for standard information criteria like sufficiency and completeness.