Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we can perform sheafification, and $X$ is a topological space (or a site if you prefer, though I don't know much about these).

The subcategory $\text{Sh}(X, \mathsf{A}) \hookrightarrow \text{PreSh}(X, \mathsf{A})$ is reflective by assumption, i.e., the embedding has sheafification as a left adjoint (called a reflector in general). Recall that a reflective subcategory is said to be exact if in addition the reflector is left exact (and hence exact).

If $\mathsf{A}$ is the category $\mathsf{Set}$, then $\text{Sh}(X, \mathsf{A})$ is an exact reflective subcategory (i.e. sheafification is exact). The embedding functor $\text{Sh}(X, \mathsf{A}) \hookrightarrow \text{PreSh}(X, \mathsf{A})$ is left exact by assumption, but I am not sure if it is right exact as well.

Now what if $\mathsf{A}$ is some other category, such as $R$-$\mathsf{Mod}$? When can we say that the sheafification and the embedding functors $\text{Sh}(X, \mathsf{A}) \hookrightarrow \text{PreSh}(X, \mathsf{A})$ are exact? How about if we are dealing with a ringed space $(X, \mathcal{O})$ and sheaves of $\mathcal{O}$-modules?

Sorry if this is a silly question; I'm studying differential geometry so I am not very familiar with sheaves yet.

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    $\begingroup$ Sheafification is exact for sheaves of abelian groups, hence also for sheaves of $R$-modules or $\mathscr{O}_X$-modules. $\endgroup$ – Zhen Lin Apr 19 '16 at 6:26
  • $\begingroup$ Thanks! How about the embedding of sheaves into presheaves? Is that exact? $\endgroup$ – ಠ_ಠ Apr 19 '16 at 7:01
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    $\begingroup$ Very rarely. That would make cohomology trivial. $\endgroup$ – Zhen Lin Apr 19 '16 at 10:37
  • $\begingroup$ Thanks! Also, is sheafification exact for sheaves of (not necessarily abelian) groups? $\endgroup$ – ಠ_ಠ Apr 19 '16 at 11:08
  • $\begingroup$ Yes, it is exact for groups. More generally, also for any finitary algebraic theory. $\endgroup$ – Zhen Lin Apr 19 '16 at 11:26

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