The Stacks project

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. $\square$

Comments (3)

Comment #1183 by JuanPablo on

I had a problem constructing the morphisms .

In the sheaf over a topological space case this morphism maps to , the problem in this proof is that if it is no longer the case that .

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

Take for example with coverings equal the jointly surjective mappings (ignoring set theoretic issues for simplicity). Take the covering where is a singleton (so a final object, and ) and is nonempty.

Then the functor of the next lemma (04TS) is equal to the restriction mapping and this lemma states that it is essentially surjective.

Then the left adjoint is its quasi-inverse, but then by the lemma 7.24.4 (tag 00Y1) the forgetful functor is an equivalence.

This arrows restrict in the embeddings from the yoneda lemma and to the forgetful functor which should then be fully faithful. If 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 . In other words, the category of glueing data is not equal to .

Comment #1185 by JuanPablo on

Ah, yes; I was confused over the ambiguity (respect to the first projection) with (respect to the second projection).

Also is a covering of over , because . So that is how is defined.

Thanks for the answer.

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