The Stacks project

Lemma 7.26.6. Let $\mathcal{C}$ be a site. The category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is equivalent to the category of absolute glueing data via the functor that associates to $\mathcal{F}$ on $\mathcal{C}$ the canonical absolute glueing data.

Proof. Given an absolute glueing data $(\mathcal{F}_ U, c_ f)$, we construct a sheaf $\mathcal{F}$ on $\mathcal{C}$ by setting $\mathcal{F}(U) = \mathcal{F}_ U(U)$, where restriction along $f : V \to U$ given by the commutative diagram

\[ \xymatrix{ \mathcal{F}_ U(U) \ar[r] \ar@{=}[d] & \mathcal{F}_ U(V) \ar[r]^{c_ f} & \mathcal{F}_ V(V) \ar@{=}[d] \\ \mathcal{F}(U) \ar[rr] & & \mathcal{F}(V) } \]

The compatibility condition $c_ g \circ j_{g}^{-1} c_ f = c_{f \circ g}$ ensures that $\mathcal{F}$ is a presheaf, and also ensures that the maps $c_ f : \mathcal{F}_ U(V) \to \mathcal{F}(V)$ define an isomorphism $\mathcal{F}_ U \to \mathcal{F}|_{\mathcal{C}/U}$ for each $U$. Since each $\mathcal{F}_ U$ is a sheaf, this implies that $\mathcal{F}$ is a sheaf as well. The functor $(\mathcal{F}_ U, c_ f) \mapsto \mathcal{F}$ just constructed is quasi-inverse to the functor which takes a sheaf on $\mathcal{C}$ to its canonical glueing data. Further details omitted. $\square$

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