A stratified directed ∞-site is an ∞-site whose objects are layered by a stratification and whose morphisms respect a directed structure. It combines three ideas: the higher categorical structure of an ∞-site (enabling homotopy-coherent sheaf conditions), the layering of objects into strata of different types or scales, and the directedness of refinement maps that prevents arbitrary reversal.
The covering families of a stratified directed ∞-site must be compatible with both the stratification and the directed structure. This means that local data lives on strata, refinements move between strata in a controlled direction, and the sheaf condition glues local data while respecting these constraints.
The stratified directed ∞-site provides a candidate geometric foundation for structures like the GFRTU, where traces are organized into layers (strata), refinement maps go from coarser to finer records, and the ∞-categorical structure supports the higher coherences needed for homotopy-type-theoretic internal logic. The (∞,1)-topos of sheaves on such a site would carry internal logic compatible with both the layered and directed aspects of the geometry.