sheafification

Let $T$ be a site. Let $P_T$ denote the category of presheaves on $T$ (with values in the category of abelian groups), and $S_T$ the category of sheaves on $T$. There is a natural inclusion functor $\iota :S_T\to P_T$.

Theorem 1

The functor $\iota$ has a left adjoint $\mathrm{♯}\mathrm{:}{P}_T\mathrm{\to }{S}_T$, that is, for any sheaf $F$ and presheaf $G$, we have

$${\mathrm{Hom}}_{S_T}(G^{\mathrm{♯}},F)\cong {\mathrm{Hom}}_{P_T}(G,\iota F).$$

This functor $\mathrm{♯}$ is called sheafification, and $G^{\mathrm{♯}}$ is called the sheafification of $F$.

One can readily check that this description in terms of adjoints characterizes $\mathrm{♯}$ completely, and that this definition reduces to the usual definition of sheafification (http://planetmath.org/Sheafification) when $T$ is the Zariski site. It also allows derivation of various exactness properties of $\mathrm{♯}$ and $\iota$.