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Remark 85.2.4 (Sheafification of presheaves of topological spaces). In this remark we briefly discuss sheafification of presheaves of topological spaces. The exact same arguments work for presheaves of topological abelian groups, topological rings, and topological modules (over a given topological ring). In order to do this in the correct generality let us work over a site $\mathcal{C}$. The reader who is interested in the case of (pre)sheaves over a topological space $X$ should think of objects of $\mathcal{C}$ as the opens of $X$, of morphisms of $\mathcal{C}$ as inclusions of opens, and of coverings in $\mathcal{C}$ as coverings in $X$, see Sites, Example 7.6.4. Denote $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top})$ the category of sheaves of topological spaces on $\mathcal{C}$ and denote $\textit{PSh}(\mathcal{C}, \textit{Top})$ the category of presheaves of topological spaces on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of topological spaces on $\mathcal{C}$. The sheafification $\mathcal{F}^\# $ should satisfy the formula

\[ \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C}, \textit{Top})}(\mathcal{F}, \mathcal{G}) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top})}(\mathcal{F}^\# , \mathcal{G}) \]

functorially in $\mathcal{G}$ from $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top})$. In other words, we are trying to construct the left adjoint to the inclusion functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top}) \to \textit{PSh}(\mathcal{C}, \textit{Top})$. We first claim that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top})$ has limits and that the inclusion functor commutes with them. Namely, given a category $\mathcal{I}$ and a functor $i \mapsto \mathcal{G}_ i$ into $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top})$ we simply define

\[ (\mathop{\mathrm{lim}}\nolimits \mathcal{G}_ i)(U) = \mathop{\mathrm{lim}}\nolimits \mathcal{G}_ i(U) \]

where we take the limit in the category of topological spaces (Topology, Lemma 5.14.1). This defines a sheaf because limits commute with limits (Categories, Lemma 4.14.10) and in particular products and equalizers (which are the operations used in the sheaf axiom). Finally, a morphism of presheaves from $\mathcal{F} \to \mathop{\mathrm{lim}}\nolimits \mathcal{G}_ i$ is clearly the same thing as a compatible system of morphisms $\mathcal{F} \to \mathcal{G}_ i$. In other words, the object $\mathop{\mathrm{lim}}\nolimits \mathcal{G}_ i$ is the limit in the category of presheaves of topological spaces and a fortiori in the category of sheaves of topological spaces. Our second claim is that any morphism of presheaves $\mathcal{F} \to \mathcal{G}$ with $\mathcal{G}$ an object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \textit{Top})$ factors through a subsheaf $\mathcal{G}' \subset \mathcal{G}$ whose size is bounded. Here we define the size $|\mathcal{H}|$ of a sheaf of topological spaces $\mathcal{H}$ to be the cardinal $\sup _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} |\mathcal{H}(U)|$. To prove our claim we let

\[ \mathcal{G}'(U) = \left\{ \quad s \in \mathcal{G}(U) \quad \middle | \quad \begin{matrix} \text{there exists a covering }\{ U_ i \to U\} _{i \in I} \\ \text{such that } s|_{U_ i} \in \mathop{\mathrm{Im}}(\mathcal{F}(U_ i) \to \mathcal{G}(U_ i)) \end{matrix} \quad \right\} \]

We endow $\mathcal{G}'(U)$ with the induced topology. Then $\mathcal{G}'$ is a sheaf of topological spaces (details omitted) and $\mathcal{G}' \to \mathcal{G}$ is a morphism through which the given map $\mathcal{F} \to \mathcal{G}$ factors. Moreover, the size of $\mathcal{G}'$ is bounded by some cardinal $\kappa $ depending only on $\mathcal{C}$ and the presheaf $\mathcal{F}$ (hint: use that coverings in $\mathcal{C}$ form a set by our conventions). Putting everything together we see that the assumptions of Categories, Theorem 4.25.3 are satisfied and we obtain sheafification as the left adjoint of the inclusion functor from sheaves to presheaves. Finally, let $p$ be a point of the site $\mathcal{C}$ given by a functor $u : \mathcal{C} \to \textit{Sets}$, see Sites, Definition 7.32.2. For a topological space $M$ the presheaf defined by the rule

\[ U \mapsto \text{Map}(u(U), M) = \prod \nolimits _{x \in u(U)} M \]

endowed with the product topology is a sheaf of topological spaces. Hence the exact same argument as given in the proof of Sites, Lemma 7.32.5 shows that $\mathcal{F}_ p = \mathcal{F}^\# _ p$, in other words, sheafification commutes with taking stalks at a point.


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