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. $\square$

Comment #1183 by JuanPablo on

I had a problem constructing the morphisms $\varphi_i : \mathcal{F}|_{\mathcal{C}/U_i} \rightarrow \mathcal{F}_i$.

In the sheaf over a topological space case this morphism maps $(s_i)$ to $s_i$, the problem in this proof is that if $V\rightarrow U_i\rightarrow U$ it is no longer the case that $V\times_U U_i=V$.

I think this lemma and the next (tag 04TS) which is based in this one are false.

Take for example $\mathcal{C}=\textit{Sets}$ with coverings equal the jointly surjective mappings (ignoring set theoretic issues for simplicity). Take the covering $A\rightarrow U$ where $U$ is a singleton (so a final object, and $\mathcal{C}/U=\mathcal{C}$) and $A$ is nonempty.

Then the functor $Sh(\mathcal{C})\rightarrow \{\text{Gluing data}\}$ of the next lemma (04TS) is equal to the restriction mapping $Sh(\mathcal{C})\rightarrow Sh(\mathcal{C}/A)$ and this lemma states that it is essentially surjective.

Then the left adjoint $Sh(\mathcal{C}/A)\rightarrow Sh(\mathcal{C})$ is its quasi-inverse, but then by the lemma 7.24.4 (tag 00Y1) the forgetful functor $Sh(\mathcal{C})/h_{A}\rightarrow Sh(\mathcal{C})$ is an equivalence.

This arrows restrict in the embeddings from the yoneda lemma $\mathcal{C}\rightarrow Sh(\mathcal{C})$ and $\mathcal{C}/A\rightarrow Sh(\mathcal{C})/h_A$ to the forgetful functor $\mathcal{C}/A\rightarrow \mathcal{C}$ which should then be fully faithful. If $A$ has more than one element it is not full.

Comment #1184 by on

OK, the counter example does not work because you have to have a glueing over $\mathcal{C}/A \times A$. In other words, the category of glueing data is not equal to $Sh(\mathcal{C}/A)$.

Comment #1185 by JuanPablo on

Ah, yes; I was confused over the ambiguity $\mathcal{F_i}\mid_{\mathcal{C}/U_i\times_U U_i}$ (respect to the first projection) with $\mathcal{F_i}\mid_{\mathcal{C}/U_i\times_U U_i}$ (respect to the second projection).

Also $\{V\times_U U_j\rightarrow V\}_j$ is a covering of $V$ over $U_i$, because $V\times_U U_j=V\times _{U_i}(U_i\times_U U_j)$. So that is how $\varphi_i(s)\mid_{V\times_U U_j/U_i}=\varphi_{ji}s_j$ is defined.

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