Definition 59.13.1. Let $\mathcal{F}$ be a presheaf on the site $\mathcal{C}$ and $\mathcal{U} = \{ U_ i \to U\} \in \text{Cov} (\mathcal{C})$. We define the *zeroth Čech cohomology group* of $\mathcal{F}$ with respect to $\mathcal{U}$ by

## 59.13 Sheafification

There is a canonical map $\mathcal{F}(U) \to \check H^0 (\mathcal{U}, \mathcal{F})$, $s \mapsto (s |_{U_ i})_{i\in I}$. We say that a *morphism of coverings* from a covering $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ to $\mathcal{U}$ is a triple $(\chi , \alpha , \chi _ j)$, where $\chi : V \to U$ is a morphism, $\alpha : J \to I$ is a map of sets, and for all $j \in J$ the morphism $\chi _ j$ fits into a commutative diagram

Given the data $\chi , \alpha , \{ \chi _ j\} _{j \in J}$ we define

We then claim that

the map is well-defined, and

depends only on $\chi $ and is independent of the choice of $\alpha , \{ \chi _ j\} _{j \in J}$.

We omit the proof of the first fact. To see part (2), consider another triple $(\psi , \beta , \psi _ j)$ with $\chi = \psi $. Then we have the commutative diagram

Given a section $s \in \mathcal{F}(\mathcal{U})$, its image in $\mathcal{F}(V_ j)$ under the map given by $(\chi , \alpha , \{ \chi _ j\} _{j \in J})$ is $\chi _ j^*s_{\alpha (j)}$, and its image under the map given by $(\psi , \beta , \{ \psi _ j\} _{j \in J})$ is $\psi _ j^*s_{\beta (j)}$. These two are equal since by assumption $s \in \check H^0(\mathcal{U}, \mathcal{F})$ and hence both are equal to the pullback of the common value

pulled back by the map $(\chi _ j, \psi _ j)$ in the diagram.

Theorem 59.13.2. Let $\mathcal{C}$ be a site and $\mathcal{F}$ a presheaf on $\mathcal{C}$.

The rule

\[ U \mapsto \mathcal{F}^+(U) := \mathop{\mathrm{colim}}\nolimits _{\mathcal{U} \text{ covering of }U} \check H^0(\mathcal{U}, \mathcal{F}) \]is a presheaf. And the colimit is a directed one.

There is a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$.

If $\mathcal{F}$ is a separated presheaf then $\mathcal{F}^+$ is a sheaf and the map in (2) is injective.

$\mathcal{F}^+$ is a separated presheaf.

$\mathcal{F}^\# = (\mathcal{F}^+)^+$ is a sheaf, and the canonical map induces a functorial isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}^\# , \mathcal{G}) \]for any $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

**Proof.**
See Sites, Theorem 7.10.10.
$\square$

In other words, this means that the natural map $\mathcal{F} \to \mathcal{F}^\# $ is a left adjoint to the forgetful functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{PSh}(\mathcal{C})$.

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