## 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 $s_ i = s|_{U_ i}$.

Remark 7.7.2. If the covering $\{ U_ i \to U\} _{i \in I}$ is the empty family (this means that $I = \emptyset$), then the sheaf condition signifies that $\mathcal{F}(U) = \{ *\}$ is a singleton set. This is because in (7.7.1.1) the second and third sets are empty products in the category of sets, which are final objects in the category of sets, hence singletons.

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.

Comment #2331 by Zili Zhang on

In Definition 7.7.5, should the description of PSh(X) be the category of presheaves of sets?

Comment #2402 by on

Not sure what the question is. The category $\textit{PSh}(\mathcal{C})$ was defined in 7.2.1.

Comment #5094 by Jodmos Horon on

"and every covering if ∅ can be refined by {∅→∅}"

Minor typo: should be "of" instead of "if".

(BTW, seems that tags got screwed up. The typo is in Example 00VP.)

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