A stratification is a decomposition of a space or category into layers called strata, ordered by refinement or dimension. Formally, a stratification of an object X assigns to each point or sub-object a stratum label from a partially ordered set, such that the strata are compatible with the object’s structure.
In topology, a stratified space is a topological space decomposed into manifold pieces of different dimensions (strata), glued together in a controlled way. The cone on a stratified space and the product of stratified spaces are again stratified. In the categorical setting, a stratification of a category C is a functor from C to a poset, partitioning objects into layers.
In a stratified directed ∞-site, the stratification organizes the objects of the ∞-site into layers, and morphisms respect this layering — they either stay within a stratum or move between strata in a controlled direction. This ensures that the covering families and sheaf conditions respect the layered structure.