A sheaf is a presheaf — an assignment of data to each object of a site — that satisfies the sheaf condition: compatible local data can be uniquely glued into global data.
A presheaf F on a category C assigns a set F(U) to each object U and a restriction map F(U) → F(V) for each morphism V → U, preserving composition and identity. When C is equipped with a Grothendieck topology (making it a site), the sheaf condition asks that for every covering family {Uᵢ → U}, the natural map
F(U) → equalizer( ∏ᵢ F(Uᵢ) ⇉ ∏ᵢ,ⱼ F(Uᵢ ×_U Uⱼ) )
is an isomorphism. This says: a global section over U is exactly a compatible family of local sections over the cover, with no ambiguity.
The prototypical example is continuous functions on a topological space: a continuous function on an open set U is determined by its values on any open cover of U, and continuous functions defined on overlapping opens that agree on their intersection glue to a continuous function on the union.
Sheaves on a site form a Grothendieck topos — a category with an internal logic (intuitionistic, not necessarily Boolean) and enough structure to do mathematics internally. The category of sheaves remembers the site’s notion of “local” through its subobject classifier and Lawvere-Tierney topology.
In the semiotic universe, the sheaf condition on fragments ensures that interpretations defined locally (on individual fragments) cohere globally — fragmentwise reasoning is valid precisely because the semiotic universe’s closure structure satisfies descent.