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Let $\mathcal{F}$ be a presheaf on a topological space $X$.

A sheafification of $\mathcal{F}$ is a sheaf $\mathcal{F}^\mathrm{sh}$ on $X$ together with a presheaf morphism $\mathrm{sh_\mathcal{F}}: \mathcal{F} \to \mathcal{F}^\mathrm{sh}$ fulfilling the universal property: For every sheaf $\mathcal{G}$ on $X$ and presheaf morphism $\phi: \mathcal{F} \to \mathcal{G}$ there is a unique sheaf morphism $\widetilde{\phi}: \mathcal{F}^\mathrm{sh} \to \mathcal{G}$ making the following diagram commute: $$ \require{AMScd} \begin{CD} \mathcal{F} @>\mathrm{sh_\mathcal{F}}>> \mathcal{F}^\mathrm{sh}\\ @VV\phi V @VV\widetilde{\phi} V\\ \mathcal{G} @>\mathrm{id}>> \mathcal{G} \end{CD} $$

Now this seams to be a "colimit type" construction and I want to know if there is a "limit type" construction. Precisely:

Let's say a sheafification of type B of $\mathcal{F}$ is a sheaf $\mathcal{F}^\mathrm{shB}$ together with a presheaf morphism $\mathrm{shB}_\mathcal{F}: \mathcal{F}^\mathrm{shB} \to \mathcal{F}$ fulfilling the universal property: For every sheaf $\mathcal{G}$ and presheaf morphism $\phi : \mathcal{G} \to \mathcal{F}^\mathrm{shB}$ there is a unique sheaf morphism $\widetilde{\phi} : \mathcal{G} \to \mathcal{F}^\mathrm{shB}$ making the following diagram commute: $$ \require{AMScd} \begin{CD} \mathcal{F} @<\mathrm{shB_\mathcal{F}}<< \mathcal{F}^\mathrm{shB}\\ @AA\phi A @AA\widetilde{\phi} A\\ \mathcal{G} @<\mathrm{id}<< \mathcal{G} \end{CD} $$

(I have basically changed the direction of all arrows from the previous definition.)

Now can anybody tell me...

  • ... whether there is an established name for that?
  • ... whether this is actually useful in (algebraic) geometry?

Also note that the left side of the diagram is always in the category of presheaves, and the right side "in the category of sheaves". Sheafification gives a left-adjoint functor to the forgetful functor $\mathrm{Shv_X} \to \mathrm{PreShv_X}$. So I guess what I am asking for is a right-adjoint functor to this forgetful functor.

Thank you so much!

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    $\begingroup$ The forgetful functor doesn’t preserve colimits, so it can’t have a right adjoint. $\endgroup$ Nov 21, 2022 at 16:29

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The question was answered in the comments: The forgetful functor from sheaves to presheaves doesn't preserve colimits, so it can't have a right adjoint. For example, it doesn't preserve coproducts, since the coproduct of a family of ($\mathbf{Set}$-valued) sheaves $(F_i)_{i \in I}$ is given by $$\left(\coprod_{i \in I} F_i\right)(U) = \left\{s \in \prod_{i \in I} F_i(U_i) : U = \coprod_{i \in I} U_i \text{ disjoint covering}\right\},$$ whereas the coproduct of presheaves is just given section-wise: $$\left(\coprod_{i \in I} F_i\right)(U) = \coprod_{i \in I} F_i(U).$$ You can already see the difference for $I = \emptyset$, i.e. the initial object. The initial sheaf on $X$ sends all non-empty open subsets of $X$ to $\emptyset$ but the empty subset to a singleton. The initial presheaf sends everything to the $\emptyset$.

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