# Direct and inverse limits of sheaves

Is the direct limit of sheaves a sheaf? Is the inverse limit of sheaves a sheaf? I guess another way of saying it is whether sheafification commutes with direct limit or inverse limit. While we are at it, does sheafification commutes with direct sum or product?

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Since no one has mentioned it, I'll just quickly say that sheafification is a left exact left adjoint, so it preserves all colimits and all finite limits. In particular it is an exact functor. –  Zhen Lin Sep 11 '12 at 1:12

Keenan Kidwell has answered your main question but I think I can add a little exposition earned from doing all the $\textrm{Ch II}\S1$ exercises in Hartshorne.

A finite (presheaf) product or coproduct of sheaves is again a sheaf, since $\mathscr{F} \oplus_{\mathsf{PSh}} \mathscr{G} \cong \mathscr{F} \times_{\mathsf{PSh}} \mathscr{G}$. Note that in general even finite colimits of sheaves have to be taken in $\mathsf{PSh}$ and then sheafified, e.g. cokernels.

However even a countably infinite family of sheaves can have the property that the direct sum(as presheaves) is not a sheaf! Take our space $\mathbb{N}_{\geq 0}$ with the discrete topology, and let $\mathscr{F}_i$ for $i \geq 0$ be the skyscraper sheaf with stalk $\mathscr{F}_{i,i}=\mathbb{Z}$ and zero elsewhere. Now we see that $\oplus_{i,\mathsf{PSh}} \mathscr{F}_i$ isn't a sheaf! Take the open sets $\{j\}_{j \geq 0}$ and the compatible family of sections $i_j(j)$ for $j \in \mathscr{F}_j(\{j\}) \cong \mathbb{Z}$ and where $i_j : \mathscr{F}_j \hookrightarrow \oplus_{i,\mathsf{PSh}} \mathscr{F}_{i}$.

EDIT: It occurs to me here that it's easier to show in general that if $\{ U_i \}$ is a family of disjoint open sets then $\mathscr{F}(\cup U_i) = \prod_i \mathscr{F}(U_i)$ if $\mathscr{F}$ is a sheaf. This immediately shows that this direct sum is not a sheaf.

It's a little dense, but if you do your part to break up that paragraph you'll see this family has no gluing, precisely because direct sums only have finitely many nonzero terms.

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It doesn't really make sense to ask "is the direct limit of sheaves a sheaf?" I assume you mean to ask whether or not the natural choice for a presheaf direct limit of a directed system of sheaves a sheaf. The answer is no in general. You have to sheafify. That is, given a directed system of sheaves $(\mathscr{F}_i,\varphi_{ij})$, the presheaf defined by $U\mapsto\varinjlim_i\mathscr{F}_i(U)$, taken with respect to the maps $\varphi_{ij}(U)$, is usually only a presheaf. The associated sheaf is what we call $\varinjlim_i\mathscr{F}$, and you can verify that it has the properties to be the (i.e. categorical) direct limit of the $\mathscr{F}_i$. I believe one case where the direct limit presheaf is already a sheaf is when $X$ is a Noetherian topological space (i.e. every subset is quasi-compact).
For an inverse system of sheaves $(\mathscr{F}_i,\varphi_{ij})$, the presheaf $U\mapsto\varprojlim_i\mathscr{F}_i(U)$ actually is a sheaf, $\varprojlim_i\mathscr{F}_i$, and it is the categorical inverse limit of the $\mathscr{F}_i$.