7.26 Glueing sheaves

This section is the analogue of Sheaves, Section 6.33.

Lemma 7.26.1. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\}$ be a covering of $\mathcal{C}$. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves on $\mathcal{C}$. Given a collection

$\varphi _ i : \mathcal{F}|_{\mathcal{C}/U_ i} \longrightarrow \mathcal{G}|_{\mathcal{C}/U_ i}$

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

$\varphi : \mathcal{F}|_{\mathcal{C}/U} \longrightarrow \mathcal{G}|_{\mathcal{C}/U}$

whose restriction to each $\mathcal{C}/U_ i$ agrees with $\varphi _ i$.

Proof. The restrictions used in the lemma are those of Lemma 7.25.8. Let $V/U$ be an object of $\mathcal{C}/U$. Set $V_ i = U_ i \times _ U V$ and denote $\mathcal{V} = \{ V_ i \to V\}$. Observe that $(U_ i \times _ U U_ j) \times _ U V = V_ i \times _ V V_ j$. Then we have $\mathcal{F}|_{\mathcal{C}/U_ i}(V_ i/U_ i) = \mathcal{F}(V_ i)$ and $\mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j}(V_ i \times _ V V_ j/U_ i \times _ U U_ j) = \mathcal{F}(V_ i \times _ V V_ j)$ and similarly for $\mathcal{G}$. Thus we can define $\varphi$ on sections over $V$ as the dotted arrows in the diagram

$\xymatrix{ \mathcal{F}(V) \ar@{=}[r] & H^0(\mathcal{V}, \mathcal{F}) \ar@{..>}[d] \ar[r] & \prod \mathcal{F}(V_ i) \ar[d]_{\prod \varphi _ i} \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \mathcal{F}(V_ i \times _ V V_ j) \ar[d]_{\prod \varphi _{ij}} \\ \mathcal{G}(V) \ar@{=}[r] & H^0(\mathcal{V}, \mathcal{G}) \ar[r] & \prod \mathcal{G}(V_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \mathcal{G}(V_ i \times _ V V_ j) }$

The equality signs come from the sheaf condition; see Section 7.10 for the notation $H^0(\mathcal{V}, -)$. We omit the verification that these maps are compatible with the restriction maps. $\square$

The previous lemma implies that given two sheaves $\mathcal{F}$, $\mathcal{G}$ on a site $\mathcal{C}$ the rule

$U \longmapsto \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}( \mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U})$

defines a sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F},\mathcal{G})$. 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 on Sites, Section 18.27.

Lemma 7.26.2. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be sheaves on $\mathcal{C}$. There is a canonical bijection

$\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F}\times \mathcal{G},\mathcal{H}) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{F},\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H}))$

which is functorial in all three entries.

Proof. The lemma says that the functors $-\times \mathcal{G}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},-)$ are adjoint to each other. To show this, we use the notion of unit and counit, see Categories, Section 4.24. The unit

$\eta _\mathcal {F} : \mathcal{F} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{F}\times \mathcal{G})$

sends $s \in \mathcal{F}(U)$ to the map $\mathcal{G}|_{\mathcal{C}/U} \to \mathcal{F}|_{\mathcal{C}/U}\times \mathcal{G}|_{\mathcal{C}/U}$ which over $V/U$ is given by

$\mathcal{G}(V) \longrightarrow \mathcal{F}(V)\times \mathcal{G}(V), \quad t \longmapsto (s|_{V},t).$

The counit

$\epsilon _{\mathcal{H}} : \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H}) \times \mathcal{G} \longrightarrow \mathcal{H}$

is the evaluation map. It is given by the rule

$\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)}( \mathcal{G}|_{\mathcal{C}/U}, \mathcal{H}|_{\mathcal{C}/U}) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (\varphi , s) \longmapsto \varphi (s).$

Then for each $\varphi : \mathcal{F} \times \mathcal{G} \to \mathcal{H}$, the corresponding morphism $\mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G},\mathcal{H})$ is given by mapping each section $s \in \mathcal{F}(U)$ to the morphism of sheaves on $\mathcal{C}/U$ which on sections over $V/U$ is given by

$\mathcal{G}(V) \longrightarrow \mathcal{H}(V),\quad t \longmapsto \varphi (s|_ V, t).$

Conversely, for each $\psi : \mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{G}, \mathcal{H})$, the corresponding morphism $\mathcal{F} \times \mathcal{G} \to \mathcal{H}$ is given by

$\mathcal{F}(U) \times \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\quad (s, t) \longmapsto \psi (s)(t)$

on sections over an object $U$. We omit the details of the proof showing that these constructions are mutually inverse. $\square$

Lemma 7.26.3. Let $\mathcal{C}$ be a site and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_ U^\# , \mathcal{F}) = j_*(\mathcal{F}|_{\mathcal{C}/U})$ for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Proof. This can be shown by directly constructing an isomorphism of sheaves. Instead we argue as follows. Let $\mathcal{G}$ be a sheaf on $\mathcal{C}$. Then

\begin{align*} \mathop{Mor}\nolimits (\mathcal{G}, j_*(\mathcal{F}|_{\mathcal{C}/U})) & = \mathop{Mor}\nolimits (\mathcal{G}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}) \\ & = \mathop{Mor}\nolimits (j_!(\mathcal{G}|_{\mathcal{C}/U}), \mathcal{F}) \\ & = \mathop{Mor}\nolimits (\mathcal{G} \times h_ U^\# , \mathcal{F}) \\ & = \mathop{Mor}\nolimits (\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_ U^\# , \mathcal{F})) \end{align*}

and we conclude by the Yoneda lemma. Here we used Lemmas 7.26.2 and 7.25.7. $\square$

Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. For each $i \in I$ let $\mathcal{F}_ i$ be a sheaf of sets on $\mathcal{C}/U_ i$. For each pair $i, j \in I$, let

$\varphi _{ij} : \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j} \longrightarrow \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j}$

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

$\xymatrix{ \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j \times _ U U_ k} \ar[rr]_{\varphi _{ik}} \ar[rd]_{\varphi _{ij}} & & \mathcal{F}_ k|_{\mathcal{C}/U_ i \times _ U U_ j \times _ U U_ k} \\ & \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j \times _ U U_ k} \ar[ru]_{\varphi _{jk}} }$

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 $\{ U_ i \to U\} _{i \in I}$.

Lemma 7.26.4. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $\{ U_ i \to U\} _{i \in I}$ there exists a sheaf of sets $\mathcal{F}$ on $\mathcal{C}/U$ together with isomorphisms

$\varphi _ i : \mathcal{F}|_{\mathcal{C}/U_ i} \to \mathcal{F}_ i$

such that the diagrams

$\xymatrix{ \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[d]_{\text{id}} \ar[r]_{\varphi _ i} & \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j} }$

are commutative.

Proof. Let us describe how to construct the sheaf $\mathcal{F}$ on $\mathcal{C}/U$. Let $a : V \to U$ be an object of $\mathcal{C}/U$. Then

$\mathcal{F}(V/U) = \{ (s_ i)_{i \in I} \in \prod _{i \in I} \mathcal{F}_ i(U_ i \times _ U V/U_ i) \mid \varphi _{ij}(s_ i|_{U_ i \times _ U U_ j \times _ U V}) = s_ j|_{U_ i \times _ U U_ j \times _ U V} \}$

We omit the construction of the restriction mappings. We omit the verification that this is a sheaf. We omit the construction of the isomorphisms $\varphi _ i$, and we omit proving the commutativity of the diagrams of the lemma. $\square$

Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}/U$. Associated to $\mathcal{F}$ we have its canonical glueing data given by the restrictions $\mathcal{F}|_{\mathcal{C}/U_ i}$ and the canonical isomorphisms

$\left(\mathcal{F}|_{\mathcal{C}/U_ i}\right)|_{\mathcal{C}/U_ i \times _ U U_ j} = \left(\mathcal{F}|_{\mathcal{C}/U_ j}\right)|_{\mathcal{C}/U_ i \times _ U U_ j}$

coming from the fact that the composition of the functors $\mathcal{C}/U_ i \times _ U U_ j \to \mathcal{C}/U_ i \to \mathcal{C}/U$ and $\mathcal{C}/U_ i \times _ U U_ j \to \mathcal{C}/U_ j \to \mathcal{C}/U$ are equal.

Lemma 7.26.5. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. The category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$ is equivalent to the category of glueing data via the functor that associates to $\mathcal{F}$ on $\mathcal{C}/U$ the canonical glueing data.

Proof. In Lemma 7.26.1 we saw that the functor is fully faithful, and in Lemma 7.26.4 we proved that it is essentially surjective (by explicitly constructing a quasi-inverse functor). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).