Restriction is the operation of moving data from a larger context to a smaller one — given a section s defined on U and an inclusion V ⊆ U, the restriction s|_V is the section obtained by “forgetting” the data outside V.
In a presheaf F on a category C, restriction is encoded by the functoriality: for each morphism i: V → U in C, there is a restriction map F(i): F(U) → F(V). These maps must satisfy:
- F(id_U) = id_{F(U)} — restricting to the same context does nothing
- F(j ∘ i) = F(i) ∘ F(j) — restricting in two steps equals restricting in one (note the contravariance)
In classical topology, if F is a presheaf of continuous functions on a space X, then restriction sends a function defined on an open set U to its restriction to a smaller open set V ⊆ U — literally evaluating the same function on fewer points.
Restriction is the “local” direction of sheaf theory: it moves from global to local. The opposite direction — assembling local data into global data — is gluing. The sheaf condition says these two directions are inverse: restriction decomposes global data into local data, and gluing reassembles it, recovering the original.
In the semiotic universe, restriction of operators to fragments is well-defined precisely for fragment-preserving operators: those whose output on a fragment stays within that fragment.