The Stacks project

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.

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 $Sh(\mathcal{C}/A)$.

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.

Thanks for the answer.