A stratum is one layer in a stratification — a single piece of the decomposition. In a stratified topological space, each stratum is typically a manifold (a smooth, uniform piece), and the full space is assembled by gluing strata of different dimensions together along their boundaries.
Strata are ordered by refinement: a lower stratum is coarser or more fundamental, a higher stratum is finer or more detailed. A refinement map from one stratum to another goes from coarser to finer, extracting more detailed information. The ordering of strata by refinement makes the collection of strata a partially ordered set (or, in the ∞-categorical setting, a poset-like structure with higher coherence).
In a stratified directed ∞-site, each object belongs to a stratum, and the directed structure constrains which morphisms are allowed between objects in different strata. This prevents arbitrary mixing of layers and ensures that the site’s covering families respect the stratification.