## 7.7 Sheaves

Let $\mathcal{C}$ be a site. Before we introduce the notion of a sheaf with values in a category we explain what it means for a presheaf of sets to be a sheaf. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$ and let $\{ U_ i \to U\} _{i\in I}$ be an element of $\text{Cov}(\mathcal{C})$. By assumption all the fibre products $U_ i \times _ U U_ j$ exist in $\mathcal{C}$. There are two natural maps

\[ \xymatrix{ \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \times _ U U_{i_1}) } \]

which we will denote $\text{pr}^*_ i$, $i = 0, 1$ as indicated in the displayed equation. Namely, an element of the left hand side corresponds to a family $(s_ i)_{i\in I}$, where each $s_ i$ is a section of $\mathcal{F}$ over $U_ i$. For each pair $(i_0, i_1) \in I \times I$ we have the projection morphisms

\[ \text{pr}^{(i_0, i_1)}_{i_0} : U_{i_0} \times _ U U_{i_1} \longrightarrow U_{i_0} \text{ and } \text{pr}^{(i_0, i_1)}_{i_1} : U_{i_0} \times _ U U_{i_1} \longrightarrow U_{i_1}. \]

Thus we may pull back either the section $s_{i_0}$ via the first of these maps or the section $s_{i_1}$ via the second. Explicitly the maps we referred to above are

\[ \text{pr}_0^* : (s_ i)_{i\in I} \longmapsto \Big( \text{pr}^{(i_0, i_1), *}_{i_0}(s_{i_0}) \Big)_{(i_0, i_1) \in I \times I} \]

and

\[ \text{pr}_1^* : (s_ i)_{i\in I} \longmapsto \Big( \text{pr}^{(i_0, i_1), *}_{i_1}(s_{i_1}) \Big)_{(i_0, i_1) \in I \times I}. \]

Finally consider the natural map

\[ \mathcal{F}(U) \longrightarrow \prod \nolimits _{i\in I} \mathcal{F}(U_ i), \quad s \longmapsto (s|_{U_ i})_{i \in I} \]

where we have used the notation $s|_{U_ i}$ to indicate the pullback of $s$ via the map $U_ i \to U$. It is clear from the functorial nature of $\mathcal{F}$ and the commutativity of the fibre product diagrams that $\text{pr}_0^*( (s|_{U_ i})_{i \in I} ) = \text{pr}_1^*( (s|_{U_ i})_{i \in I} )$.

Definition 7.7.1. Let $\mathcal{C}$ be a site, and let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. We say $\mathcal{F}$ is a *sheaf* if for every covering $\{ U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})$ the diagram

7.7.1.1
\begin{equation} \label{sites-equation-sheaf-condition} \xymatrix{ \mathcal{F}(U) \ar[r] & \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \times _ U U_{i_1}) } \end{equation}

represents the first arrow as the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$.

Loosely speaking this means that given sections $s_ i \in \mathcal{F}(U_ i)$ such that

\[ s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j} \]

in $\mathcal{F}(U_ i \times _ U U_ j)$ for all pairs $(i, j) \in I \times I$ then there exists a unique $s \in \mathcal{F}(U)$ such that $s_ i = s|_{U_ i}$.

Example 7.7.3. Let $X$ be a topological space. Let $X_{Zar}$ be the site constructed in Example 7.6.4. The notion of a sheaf on $X_{Zar}$ coincides with the notion of a sheaf on $X$ introduced in Sheaves, Definition 6.7.1.

Example 7.7.4. Let $X$ be a topological space. Let us consider the site $X'_{Zar}$ which is the same as the site $X_{Zar}$ of Example 7.6.4 except that we disallow the empty covering of the empty set. In other words, we do allow the covering $\{ \emptyset \to \emptyset \} $ but we do not allow the covering whose index set is empty. It is easy to show that this still defines a site. However, we claim that the sheaves on $X'_{Zar}$ are different from the sheaves on $X_{Zar}$. For example, as an extreme case consider the situation where $X = \{ p\} $ is a singleton. Then the objects of $X'_{Zar}$ are $\emptyset , X$ and every covering of $\emptyset $ can be refined by $\{ \emptyset \to \emptyset \} $ and every covering of $X$ by $\{ X \to X\} $. Clearly, a sheaf on this is given by any choice of a set $\mathcal{F}(\emptyset )$ and any choice of a set $\mathcal{F}(X)$, together with any restriction map $\mathcal{F}(X) \to \mathcal{F}(\emptyset )$. Thus sheaves on $X'_{Zar}$ are the same as usual sheaves on the two point space $\{ \eta , p\} $ with open sets $\{ \emptyset , \{ \eta \} , \{ p, \eta \} \} $. In general sheaves on $X'_{Zar}$ are the same as sheaves on the space $X \amalg \{ \eta \} $, with opens given by the empty set and any set of the form $U \cup \{ \eta \} $ for $U \subset X$ open.

Definition 7.7.5. The category *$\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$* of sheaves of sets is the full subcategory of the category $\textit{PSh}(\mathcal{C})$ whose objects are the sheaves of sets.

Let $\mathcal{A}$ be a category. If products indexed by $I$, and $I \times I$ exist in $\mathcal{A}$ for any $I$ that occurs as an index set for covering families then Definition 7.7.1 above makes sense, and defines a notion of a sheaf on $\mathcal{C}$ with values in $\mathcal{A}$. Note that the diagram in $\mathcal{A}$

\[ \xymatrix{ \mathcal{F}(U) \ar[r] & \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \times _ U U_{i_1}) } \]

is an equalizer diagram if and only if for every object $X$ of $\mathcal{A}$ the diagram of sets

\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(X, \mathcal{F}(U)) \ar[r] & \prod \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(X, \mathcal{F}(U_ i)) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(X, \mathcal{F}(U_{i_0} \times _ U U_{i_1})) } \]

is an equalizer diagram.

Suppose $\mathcal{A}$ is arbitrary. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{A}$. Choose any object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. Then we get a presheaf of sets $\mathcal{F}_ X$ defined by the rule

\[ \mathcal{F}_ X(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(X, \mathcal{F}(U)). \]

From the above it follows that a good definition is obtained by requiring all the presheaves $\mathcal{F}_ X$ to be sheaves of sets.

Definition 7.7.6. Let $\mathcal{C}$ be a site, let $\mathcal{A}$ be a category and let $\mathcal{F}$ be a presheaf on $\mathcal{C}$ with values in $\mathcal{A}$. We say that $\mathcal{F}$ is a *sheaf* if for all objects $X$ of $\mathcal{A}$ the presheaf of sets $\mathcal{F}_ X$ (defined above) is a sheaf.

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