Functorial isomorphisms

Question

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\PSh{PSh}$We know that a presheaf $\mathcal{F}$ on category $ \mathcal{C} $ gives a colimit preasheaf $ \mathcal{F}^{+} $ defined by Čech cohomology, and then we have the following theorem: $ \mathcal{F}^{\sharp} = (\mathcal{F}^{+})^{+} $ is a sheaf and the canonical map induces a functorial isomorphism $ \Hom_{\PSh(\mathcal{C})}(\mathcal{F} , \mathcal{G}) = \Hom_{\Sh(\mathcal{C})}(\mathcal{F}^{\sharp} , \mathcal{G}) $, for all $ \mathcal{G} $ in $ \Sh(\mathcal{C}) $.

My question is, can we can find a functorial isomorphism between $ \Hom_{\PSh(\mathcal{C})}(\mathcal{F}^{+} , \mathcal{G}) $ and $ \Hom_{\Sh(\mathcal{C})}(\mathcal{F}^{\sharp} , \mathcal{G}) $?

And the analogous question for $ \Hom_{\PSh(\mathcal{C})}(\mathcal{F} , \mathcal{G}) $ and $ \Hom_{\Sh(\mathcal{C})}((\mathcal{F}^{\sharp})^{+} , \mathcal{G}) $.

Answer 1

Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space. More generally, we can assume $C$ to be an arbitrary site.

In this case, the answer to both questions is yes. It follows from a more general result:

If $G$ is a sheaf and $f\colon P\to Q$ is a local isomorphism of presheaves (i.e., a morphism that becomes an isomorphism after passing to associated sheaves; in the case of sites with enough points, such as topological spaces, it can be described as a morphism that induces an isomorphism on all stalks), then the induced map $$\def\Hom{\mathop{\rm Hom}} \Hom(Q,G)→\Hom(P,G)$$ is an isomorphism.

This abstract result applies to the two cases under consideration because the maps $F^+→F^\sharp$ and $F\to (F^\sharp)^+=F^\sharp$ are local isomorphisms.

Indeed, even more generally, the natural map $F\to F^+$ is a local isomorphism for any presheaf $F$. Indeed, the associated sheaf functor can be computed as $F↦F^{++}$, so if we apply it to the morphism $F→F^+$, we get the identity map $F^{++}→F^{+++}=F^{++}$.

In the case of sites with enough points, this also follows immediately from the fact that the stalk functor is cocontinuous, in particular, it preserves the colimit used to define $F^+$.