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This is probably an extremely elementary question, and I am not sure if it is down to my lack of understanding of sheafification, or my lack of understanding of what it means for a right (respectively left) adjoint to preserve limits (respectively colimits).

Suppose I have sheaves of abelian groups $\mathcal{F}$ and $\mathcal{G}$. Say I want to show that the presheaf defined on open sets $U$ by $U \mapsto \mathcal{F}(U) \oplus \mathcal{G}(U)$ is itself a sheaf. I would like to do this using adjointness properties of sheafification and the forgetful functor. However, it seems to work the opposite to what I would expect. Is it correct that all I need to do is sheafify this presheaf, show that the resulting sheaf satisfies the universal property for direct sum (hence is a limit) in the category of sheaves, then apply the forgetful functor and claim that the result must be the original prehseaf?

My problem is that I know in general that limits play nice with sheafificaioin, not colimits. So say I had a family $\{ \mathcal{F}_{i} \}_{i \in I}$ of sheaves. Now suppose I defined a presheaf via the colimit $$ U \mapsto \lim_{\rightarrow} \mathcal{F}_{i}(U) $$ Since sheafification is a left adjoint, it preserves colimits. So why isn't the sheafificaton of this presheaf colimit simply the sheaf colimit, which would imply it is a sheaf - a 'fact" I know to be false. I seem to be understanding "preserves (co)limit" as the inverse of what it actually means in some sense. Why do we use the forgetful functor properties and not sheafification?

I apologize if this is a poorly stated question, but I have some vague understanding, but I feel like I'm just not seeing it properly. Any explanation of these examples, or really any general comments would be appreciated.

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  • $\begingroup$ You should absolutely go through the exercise of proving that right/left adjoints preserve limits/colimits. You should prove it for various formulations of adjunction and (co)limits. This is a fundamental fact with which you should become intimately familiar. $\endgroup$ – Derek Elkins left SE Aug 7 '17 at 2:56
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Sheafification preserves colimits: that means that you can construct a colimit of sheaves by constructing the same colimit in presheaves, and then sheafifying. But this is not what you want to show in your example. The claim there is that the direct sum of presheaves is already a sheaf, without sheafifying, a subtlely but critically different claim.

If you want to use adjointness properties for this, then you need to observe that a direct sum is also a limit and prove that limits of sheaves (among presheaves) are still sheaves. If you assume that limits of sheaves exist, then this follows from the inclusion of sheaves into presheaves being right adjoint to sheafification. But to prove that such limits exist is a slightly tricky exercise-the general result is to prove that any fully faithful right adjoint creates limits, which is certainly worth proving or looking up.

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