The Stacks project

Colimit in category of sheaves equals sheafification of colimit in category of presheaves.

Lemma 7.10.13. Let $\mathcal{F} : \mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a diagram. Then $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} \mathcal{F}$ exists and is the sheafification of the colimit in the category of presheaves.

Proof. Since the sheafification functor is a left adjoint it commutes with all colimits, see Categories, Lemma 4.24.5. Hence, since $\textit{PSh}(\mathcal{C})$ has colimits, we deduce that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ has colimits (which are the sheafifications of the colimits in presheaves). $\square$

Comments (1)

Comment #7344 by Alejandro González Nevado on

Slogan suggestion (from a sentence used in the proof of Lemma 00XK): Colimit in category of sheaves equals sheafification of colimit in category of presheaves.

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  • 8 comment(s) on Section 7.10: Sheafification

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