## 6.33 Glueing sheaves

In this section we glue sheaves defined on the members of a covering of $X$. We first deal with maps.

Lemma 6.33.1. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of sets on $X$. Given a collection

\[ \varphi _ i : \mathcal{F}|_{U_ i} \longrightarrow \mathcal{G}|_{U_ i} \]

of maps of sheaves such that for all $i, j \in I$ the maps $\varphi _ i, \varphi _ j$ restrict to the same map $\mathcal{F}|_{U_ i \cap U_ j} \to \mathcal{G}|_{U_ i \cap U_ j}$ then there exists a unique map of sheaves

\[ \varphi : \mathcal{F} \longrightarrow \mathcal{G} \]

whose restriction to each $U_ i$ agrees with $\varphi _ i$.

**Proof.**
For each open subset $U \subset X$ define

\[ \varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U), \quad s \mapsto \varphi _ U(s) \]

where $\varphi _ U(s)$ is the unique section verifying

\[ (\varphi _ U(s))|_{U \cap U_ i} = (\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}). \]

Existence and uniqueness of such a section follows from the sheaf axioms due to the fact that

\begin{align*} ((\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}))|_{U \cap U_ i \cap U_ j} & = (\varphi _ i)_{U \cap U_ i \cap U_ j}(s|_{U \cap U_ i \cap U_ j})\\ & = (\varphi _ j)_{U \cap U_ i \cap U_ j}(s|_{U \cap U_ i \cap U_ j})\\ & = ((\varphi _ j)_{U \cap U_ j}(s|_{U \cap U_ j}))|_{U \cap U_ i \cap U_ j}. \end{align*}

This family of maps gives us indeed a map of sheaves: Let $V \subset U \subset X$ be open subsets then

\[ (\varphi _ U(s))|_ V = \varphi _ V(s|_ V) \]

since for each $i \in I$ the following holds

\begin{align*} (\varphi _ U(s))|_{V \cap U_ i} & = ((\varphi _ U(s))|_{U \cap U_ i})|_{V \cap U_ i}\\ & = ((\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}))|_{V \cap U_ i}\\ & = (\varphi _ i)_{V \cap U_ i}(s|_{V \cap U_ i})\\ & = \varphi _ V(s_{V})|_{V \cap U_ i}. \end{align*}

Furthermore, its restriction to each $U_ i$ agrees with $\varphi _ i$ since given $U \subset X$ open subset and $s \in \mathcal{F}(U \cap U_ i)$ then

\begin{align*} \varphi _{U \cap U_ i}(s) & = \varphi _{U \cap U_ i}(s)|_{U \cap U_ i}\\ & = (\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i})\\ & = (\varphi _ i)_{U \cap U_ i}(s). \end{align*}

$\square$
The previous lemma implies that given two sheaves $\mathcal{F}$, $\mathcal{G}$ on the topological space $X$ the rule

\[ U \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}( \mathcal{F}|_ U, \mathcal{G}|_ U) \]

defines a sheaf. This is a kind of *internal hom sheaf*. It is seldom used in the setting of sheaves of sets, and more usually in the setting of sheaves of modules, see Modules, Section 17.22.

Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. For each $i \in I$ let $\mathcal{F}_ i$ be a sheaf of sets on $U_ i$. For each pair $i, j \in I$, let

\[ \varphi _{ij} : \mathcal{F}_ i|_{U_ i \cap U_ j} \longrightarrow \mathcal{F}_ j|_{U_ i \cap U_ j} \]

be an isomorphism of sheaves of sets. Assume in addition that for every triple of indices $i, j, k \in I$ the following diagram is commutative

\[ \xymatrix{ \mathcal{F}_ i|_{U_ i \cap U_ j \cap U_ k} \ar[rr]_{\varphi _{ik}} \ar[rd]_{\varphi _{ij}} & & \mathcal{F}_ k|_{U_ i \cap U_ j \cap U_ k} \\ & \mathcal{F}_ j|_{U_ i \cap U_ j \cap U_ k} \ar[ru]_{\varphi _{jk}} } \]

We will call such a collection of data $(\mathcal{F}_ i, \varphi _{ij})$ a *glueing data for sheaves of sets with respect to the covering $X = \bigcup U_ i$*.

Lemma 6.33.2. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $X = \bigcup U_ i$ there exists a sheaf of sets $\mathcal{F}$ on $X$ together with isomorphisms

\[ \varphi _ i : \mathcal{F}|_{U_ i} \to \mathcal{F}_ i \]

such that the diagrams

\[ \xymatrix{ \mathcal{F}|_{U_ i \cap U_ j} \ar[r]_{\varphi _ i} \ar[d]_{\text{id}} & \mathcal{F}_ i|_{U_ i \cap U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}|_{U_ i \cap U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j|_{U_ i \cap U_ j} } \]

are commutative.

**Proof.**
First proof. In this proof we give a formula for the set of sections of $\mathcal{F}$ over an open $W \subset X$. Namely, we define

\[ \mathcal{F}(W) = \{ (s_ i)_{i \in I} \mid s_ i \in \mathcal{F}_ i(W \cap U_ i), \varphi _{ij}(s_ i|_{W \cap U_ i \cap U_ j}) = s_ j|_{W \cap U_ i \cap U_ j} \} . \]

Restriction mappings for $W' \subset W$ are defined by the restricting each of the $s_ i$ to $W' \cap U_ i$. The sheaf condition for $\mathcal{F}$ follows immediately from the sheaf condition for each of the $\mathcal{F}_ i$.

We still have to prove that $\mathcal{F}|_{U_ i}$ maps isomorphically to $\mathcal{F}_ i$. Let $W \subset U_ i$. In this case the condition in the definition of $\mathcal{F}(W)$ implies that $s_ j = \varphi _{ij}(s_ i|_{W \cap U_ j})$. And the commutativity of the diagrams in the definition of a glueing data assures that we may start with *any* section $s \in \mathcal{F}_ i(W)$ and obtain a compatible collection by setting $s_ i = s$ and $s_ j = \varphi _{ij}(s_ i|_{W \cap U_ j})$.

Second proof (sketch). Let $\mathcal{B}$ be the set of opens $U \subset X$ such that $U \subset U_ i$ for some $i \in I$. Then $\mathcal{B}$ is a base for the topology on $X$. For $U \in \mathcal{B}$ we pick $i \in I$ with $U \subset U_ i$ and we set $\mathcal{F}(U) = \mathcal{F}_ i(U)$. Using the isomorphisms $\varphi _{ij}$ we see that this prescription is “independent of the choice of $i$”. Using the restriction mappings of $\mathcal{F}_ i$ we find that $\mathcal{F}$ is a sheaf on $\mathcal{B}$. Finally, use Lemma 6.30.6 to extend $\mathcal{F}$ to a unique sheaf $\mathcal{F}$ on $X$.
$\square$

Lemma 6.33.3. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $(\mathcal{F}_ i, \varphi _{ij})$ be a glueing data of sheaves of abelian groups, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules for some sheaf of rings $\mathcal{O}$ on $X$. Then the construction in the proof of Lemma 6.33.2 above leads to a sheaf of abelian groups, resp. sheaf of algebraic structures, resp. sheaf of $\mathcal{O}$-modules.

**Proof.**
This is true because in the construction the set of sections $\mathcal{F}(W)$ over an open $W$ is given as the equalizer of the maps

\[ \xymatrix{ \prod _{i \in I} \mathcal{F}_ i(W \cap U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j\in I} \mathcal{F}_ i(W \cap U_ i \cap U_ j) } \]

And in each of the cases envisioned this equalizer gives an object in the relevant category whose underlying set is the object considered in the cited lemma.
$\square$

Lemma 6.33.4. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. The functor which associates to a sheaf of sets $\mathcal{F}$ the following collection of glueing data

\[ (\mathcal{F}|_{U_ i}, (\mathcal{F}|_{U_ i})|_{U_ i \cap U_ j} \to (\mathcal{F}|_{U_ j})|_{U_ i \cap U_ j} ) \]

with respect to the covering $X = \bigcup U_ i$ defines an equivalence of categories between $\mathop{\mathit{Sh}}\nolimits (X)$ and the category of glueing data. A similar statement holds for abelian sheaves, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules.

**Proof.**
The functor is fully faithful by Lemma 6.33.1 and essentially surjective (via an explicitly given quasi-inverse functor) by Lemma 6.33.2.
$\square$

This lemma means that if the sheaf $\mathcal{F}$ was constructed from the glueing data $(\mathcal{F}_ i, \varphi _{ij})$ and if $\mathcal{G}$ is a sheaf on $X$, then a morphism $f : \mathcal{F} \to \mathcal{G}$ is given by a collection of morphisms of sheaves

\[ f_ i : \mathcal{F}_ i \longrightarrow \mathcal{G}|_{U_ i} \]

compatible with the glueing maps $\varphi _{ij}$. Similarly, to give a morphism of sheaves $g : \mathcal{G} \to \mathcal{F}$ is the same as giving a collection of morphisms of sheaves

\[ g_ i : \mathcal{G}|_{U_ i} \longrightarrow \mathcal{F}_ i \]

compatible with the glueing maps $\varphi _{ij}$.

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