An ∞-site is an ∞-category equipped with a Grothendieck topology — a specification of covering families for each object, satisfying higher-categorical analogues of the axioms for a site. The Grothendieck topology on an ∞-category determines which presheaves (∞-functors to spaces) satisfy the sheaf condition, now understood as homotopy descent.
The category of sheaves on an ∞-site is an (∞,1)-topos — the higher-categorical analogue of a Grothendieck topos. The objects are sheaves of spaces (∞-sheaves), and the sheaf condition requires that compatible local data glues uniquely up to coherent homotopy. This is the natural setting for homotopy-coherent geometry and logic.
In a stratified directed ∞-site, the ∞-site structure is combined with stratification (layers) and directedness (irreversible morphisms). The resulting structure supports sheaf conditions that respect both the layering and the directional constraints — covering families must be compatible with strata and with the preferred direction of morphisms.