The Stacks project

Lemma 6.30.9. Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $\mathcal{B}$. There exists a unique sheaf $\mathcal{F}^{ext}$ with values in $\mathcal{C}$ on $X$ such that $\mathcal{F}^{ext}(U) = \mathcal{F}(U)$ for all $U \in \mathcal{B}$ compatibly with the restriction mappings.

Proof. By the conditions imposed on the pair $(\mathcal{C}, F)$ it suffices to come up with a presheaf $\mathcal{F}^{ext}$ which does the correct thing on the level of underlying presheaves of sets. Thus our first task is to construct a suitable object $\mathcal{F}^{ext}(U)$ for all open $U \subset X$. We could do this by imitating Lemma 6.18.1 in the setting of presheaves on $\mathcal{B}$. However, a slightly different method (but basically equivalent) is the following: Define it as the directed colimit

\[ \mathcal{F}^{ext}(U) := \mathop{\mathrm{colim}}\nolimits _\mathcal {U} FIB(\mathcal{U}) \]

over all coverings $\mathcal{U} : U = \bigcup _{i\in I} U_ i$ by $U_ i \in \mathcal{B}$ of the fibre product

\[ \xymatrix{ FIB(\mathcal{U}) \ar[r] \ar[d] & \prod \nolimits _{x\in U} \mathcal{F}_ x \ar[d] \\ \prod \nolimits _{i\in I} \mathcal{F}(U_ i) \ar[r] & \prod \nolimits _{i \in I} \prod \nolimits _{x\in U_ i} \mathcal{F}_ x } \]

By the usual arguments, see Lemma 6.15.4 and Example 6.15.5 it suffices to show that this construction on underlying sets is the same as the definition using $(**)$ above. Details left to the reader. $\square$

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