A site is a category C equipped with a Grothendieck topology — a specification of which families of morphisms into an object count as covers. A site generalizes the notion of a topological space: instead of open sets and their inclusions, a site has objects and covering families, and the covering families determine what “local” means.
A Grothendieck topology J on C assigns to each object U a collection J(U) of covering sieves (or covering families) satisfying three axioms: the maximal sieve covers (the identity covers every object), stability under pullback (pulling back a cover along any morphism gives a cover), and transitivity (a cover of a cover is a cover). These axioms ensure that the notion of “local data” is well behaved.
A presheaf on a site is a functor F: Cᵒᵖ → Set. A sheaf is a presheaf satisfying the sheaf condition with respect to the Grothendieck topology: compatible local data glues uniquely. The category of sheaves on a site is a Grothendieck topos — the generalization of the category of sheaves on a topological space. In the Interactive Semioverse, the poset of fragments under inclusion carries a covering structure, and the sheaf condition ensures that local semantic data assembles coherently.