 Direct sum of sheaves

The direct sum of sheaves of abelian groups is the sheafification of the presheaf $U\mapsto {⨁}_i{\mathcal{F}}_i(U)$. It need not equal that presheaf: sections are locally represented using finitely many summands, while globally infinitely many can occur. For point-supported skyscraper sheaves at closed points, allowable point supports are a locally finite family of subsets.