  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.
