A directed structure on a category or ∞-category means that morphisms have a preferred direction: not all morphisms are invertible. In an ordinary category, this is the default — morphisms go from source to target and need not have inverses. The term “directed” is emphasized when contrasting with groupoids or ∞-groupoids, where all morphisms are invertible.
In directed homotopy theory, paths have a preferred direction and cannot be freely reversed. The fundamental category of a directed space replaces the fundamental group of classical homotopy theory, capturing asymmetric connectivity. This directedness models irreversible processes — state transitions, time evolution, causal order.
In a stratified directed ∞-site, the directed structure constrains how morphisms move between strata: refinement maps go from coarser to finer layers, and this directedness is not invertible. The combination of directedness with stratification and higher categorical structure produces a site suited for modeling processes that are both layered and irreversible.