The Stacks project

Remark 7.26.7. There is a variant of Lemma 7.26.6 which comes up in algebraic geometry. Namely, suppose that $\mathcal{C}$ is a site with all fibre products and for each $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we are given a full subcategory $U_\tau \subset \mathcal{C}/U$ with the following properties

  1. $U/U$ is in $U_\tau $,

  2. for $V/U$ in $U_\tau $ and covering $\{ V_ j \to V\} $ of $\mathcal{C}$ we have $V_ j/U$ in $U_\tau $ and

  3. for a morphism $U' \to U$ of $\mathcal{C}$ and $V/U$ in $U_\tau $ the base change $V' = V \times _ U U'$ is in $U'_\tau $.

In this setting $U_\tau $ is a site for all $U$ in $\mathcal{C}$ and the base change functor $U_\tau \to U'_\tau $ defines a morphism $f_\tau : U_\tau \to U'_\tau $ of sites for all morphisms $f : U' \to U$ of $\mathcal{C}$. The glueing statement we obtain then reads as follows: A sheaf $\mathcal{F}$ on $\mathcal{C}$ is given by the following data:

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a sheaf $\mathcal{F}_ U$ on $U_\tau $,

  2. for every $f : U' \to U$ in $\mathcal{C}$ a map $c_ f : f_\tau ^{-1}\mathcal{F}_ U \to \mathcal{F}_{U'}$.

These data are subject to the following conditions:

  1. given $f : U' \to U$ and $g : U'' \to U'$ in $\mathcal{C}$ the composition $c_ g \circ g_\tau ^{-1}c_ f$ is equal to $c_{f \circ g}$, and

  2. if $f : U' \to U$ is in $U_\tau $ then $c_ f$ is an isomorphism.

If we ever need this we will precisely state and prove this here. (Note that this result is slightly different from the statements above as we are not requiring all the maps $c_ f$ to be isomorphisms!)

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