Sheaf on a site

Remarks

Equivalently, $P$ is a sheaf when, for every covering sieve $S \subseteq \Hom(-, c)$, every natural transformation $S \to P$ extends uniquely to $\Hom(-, c) \to P$. A morphism of sheaves is a natural transformation; sheaves on $(\mathcal{C}, J)$ form a full subcategory $\mathrm{Sh}(\mathcal{C}, J)$ of presheaves. Example: on the site of open sets of a space $X$, the presheaf $P(U) = C(U, \R)$ of continuous functions is a sheaf. Functions on the members of a covering sieve that agree on overlaps define one function on $U$, which is continuous because continuity is local. Non-example: on the same site, the constant presheaf $P(U) = \set{0, 1}$ fails, because the empty cover of $U = \emptyset$ has a unique (empty) matching family but two amalgamations. For this site the definition gives back the sheaves on a topological space.