Sheafification

Remarks

A sheafification exists for every presheaf on a small site and is unique up to unique isomorphism. So $P \mapsto aP$ is a functor $\mathrm{PSh}(\mathcal{C}) \to \mathrm{Sh}(\mathcal{C}, J)$, left adjoint to the inclusion, and it preserves finite limits. One construction is $aP = (P^+)^+$, where $P^+(c)$ is the colimit over covering sieves $S \in J(c)$ of the sets of matching families for $S$. If $P$ is already a sheaf, then $\eta$ is an isomorphism. Example: on the site of open sets of a space $X$, the sheafification of the constant presheaf $U \mapsto A$ is the sheaf of locally constant functions $U \to A$. Non-example: let $A = \set{0, 1}$. The constant presheaf itself, with $\eta = \id$, is not a sheafification, since it is not a sheaf. Indeed, the empty sieve covers $\emptyset$, and its unique (empty) matching family has two amalgamations in $P(\emptyset) = A$. The sheafification repairs this: the sheaf of locally constant functions has exactly one section over $\emptyset$.