Is the limit of a family of sheaves a sheaf?

Question

So, I can prove that the kernel of a morphism of sheaves or a product of sheaves is a sheaf, but I do not know how to prove in general that $lim F_{i}$ is a sheaf for $F_{i}$ sheaves. I know that if we regard the $F_{i}$ as presheaves, then the limit exists in the category of presheaves but I do not know how to argue about the category of sheaves.

Answer 1

Recall we have an adjoint pair $a \dashv i$, where $i : Sh(X) \to PSh(X)$ is the inclusion of sheaves into presheaves, and $a : PSh(X) \to Sh(X)$ is the sheafification functor.

Now since right adjoints preserve limits we can compute the limit of a diagram in $Sh(X)$ as though it were a diagram in $PSh(X)$. That is, if $D$ is a diagram in $Sh(X)$, then computing $\lim_D \mathcal{F}_\alpha$ as though the $\mathcal{F}_\alpha$s were merely presheaves will always result in a sheaf.

If you want to be super precise, this is because

$$ \lim_D i \mathcal{F}_\alpha \cong i \left ( \lim_D \mathcal{F}_\alpha \right ) $$

Notice, however, that right adjoints do not preserve colimits. So $\text{colim}_D \mathcal{F}_\alpha$ need not be a sheaf if we interpret the $\mathcal{F}_\alpha$ as presheaves! We solve this problem by sheafifying afterwards:

$$\text{colim}_D \mathcal{F}_\alpha \cong a \left ( \text{colim}_D i \mathcal{F}_\alpha \right )$$

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For concreteness, notice this is analogous to the situation of abelian groups sitting inside of groups.

We have an adjoint pair $ab \dashv i$, where $ab(G)$ is the abelianization functor, and $i$ is the inclusion of abelian groups into groups.

Then, since right adjoints preserve limits, the limit of abelian groups is exactly the same as the limit computed as though they were merely groups: $A \times B$ means the same thing whether $A$ and $B$ are abelian, or whether they are just groups.

However, the colimit of abelian groups is not the same thing as the colimit of groups!

The colimit $A \ast B$ computed in $\mathsf{Grp}$ is the free product, whereas the colimit $A \oplus B$ as computed in $\mathsf{Ab}$ is the direct sum! But notice that the direct sum is the abelianization of the free product! That is, $A \oplus B \cong ab \left ( A \ast B \right )$.

The situation is entirely analogous for sheaves as a subcategory of presheaves.

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I hope this helps ^_^

Answer 2

A proof by HallaSurvivor was given using an existence of a sheafification. That is, of a left adjoint to the inclusion $\mathsf{PSh}(X)\to\mathsf{PSh}(X)$ (so $\mathsf{Sh}(X)$ is a reflective subcategory of $\mathsf{PSh}(X)$). By theory of reflective subcategories it follows that the inclusion creates limits. I will here show that the inclusion strictly creates object-wise computable limits, without invoking any sheafification. (The expressions "creates limits" and "strictly creates limits" are explained in definitions 3.3.1 and 3.3.7 of Riehl's book).

Two important facts must be used. Let $\mathsf{I}, \mathsf{J}$ be a small categories.

1. A diagram of functors $\{F_i:\mathsf{C}\to\mathsf{D}\}_{i\in \mathsf{J}}$ (that is, a diagram $\mathsf{J}\to\mathsf{D}^\mathsf{C}$) which has object-wise limit (i.e., for each $c\in\mathsf{C}$, the limit $\lim_{\mathsf{J}}F_i(c)$ exists in $\mathsf{D}$) also has limit and it is computable object-wise. That is, $\lim_\mathsf{J}F_i$ exists in $\mathsf{D}^\mathsf{C}$ and $(\lim_\mathsf{J}F_i)(c)\cong\lim_\mathsf{J}F_i(c)$, for each $c\in\mathsf{C}$. You can read about this fact here and also in Proposition 3.3.9 of Riehl's book.

2. Limits commute with limits: if we have a diagram $\{c_{ij}\}_{\mathsf{I}\times\mathsf{J}}$ in $\mathsf{C}$ (i.e., a diagram $\mathsf{I}\times\mathsf{J}\to\mathsf{C}$) for which $\lim_{j\in\mathsf{J}}c_{ij}$ exists for all $i\in\mathsf{I}$, then $\lim_{i\in\mathsf{I}}\lim_{j\in\mathsf{J}}c_{ij}$ exists if and only if $\lim_{(i,j)\in\mathsf{I}\times\mathsf{J}}c_{ij}$ exists, and in this case $$ \lim_{i\in\mathsf{I}}\lim_{j\in\mathsf{J}}c_{ij}\cong\lim_{(i,j)\in\mathsf{I}\times\mathsf{J}}c_{ij}. $$ The same holds if we swap $\mathsf{I}$ and $\mathsf{J}$. Thus, if all the involved limits exist, $$ \lim_{i\in\mathsf{I}}\lim_{j\in\mathsf{J}}c_{ij}\cong\lim_{j\in\mathsf{J}}\lim_{i\in\mathsf{I}}c_{ij}. $$ The analogous result holds for colimits. A quick search in google with "limits commute with limits in category theory" should give explanations of this fact.

We are ready to give our proof. Suppose $\{\mathcal{F}_i\}_{i\in\mathsf{J}}$ is a diagram of sheaves (a functor $\mathsf{J}\to\mathsf{Sh}(X)$) that has a limit in $\mathsf{PSh}(X)$. For the proof, we need to suppose that the diagram $\{\mathcal{F}_i\}_{i\in\mathsf{J}}$ has object-wise limits. That is, for each $U\in\operatorname{Ouv}(X)$, we assume that $\lim_i\mathcal{F}(U)$ exists in the category where our sheaves takes values. This would be true were this category complete, as it is for the usual categories ($\mathsf{Set},\mathsf{Ring},\mathsf{Mod}_R,\mathsf{Ab},\dots$). Then, for any open set $U\subset X$ and any open cover $\bigcup_\alpha U_\alpha=U$, we have

\begin{aligned} &\phantom{\cong}\operatorname{eq}\left( \prod_\alpha(\lim_i\mathcal{F}_i)(U_\alpha)\rightrightarrows \prod_{\alpha,\beta}(\lim_i\mathcal{F}_i)(U_\alpha\cap U_\beta) \right)\cong\\ &\cong\operatorname{eq}\left( \prod_\alpha\lim_i\mathcal{F}_i(U_\alpha)\rightrightarrows \prod_{\alpha,\beta}\lim_i\mathcal{F}_i(U_\alpha\cap U_\beta) \right) &&\text{by fact 1}\\ &\cong\lim_i\operatorname{eq}\left( \prod_\alpha\mathcal{F}_i(U_\alpha)\rightrightarrows \prod_{\alpha,\beta}\mathcal{F}_i(U_\alpha\cap U_\beta) \right)&&\text{by fact 2}\\ &\cong\lim_i\mathcal{F}_i(U)&\text{since }\mathcal{F}_i\text{ are sheaves}\\ &\cong(\lim_i\mathcal{F}_i)(U)&&\text{by fact 1}. \end{aligned} So $\lim_i\mathcal{F}_i$ satisfies the sheaf condition.