The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.4 Limits and colimits of presheaves

Let $\mathcal{C}$ be a category. Limits and colimits exist in the category $\textit{PSh}(\mathcal{C})$. In addition, for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the functor

\[ \textit{PSh}(\mathcal{C}) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}(U) \]

commutes with limits and colimits. Perhaps the easiest way to prove these statements is the following. Given a diagram $ \mathcal{F} : \mathcal{I} \to \textit{PSh}(\mathcal{C}) $ define presheaves

\[ \mathcal{F}_{\mathop{\mathrm{lim}}\nolimits } : U \longmapsto \mathop{\mathrm{lim}}\nolimits _{i \in \mathcal{I}} \mathcal{F}_ i(U) \text{ and } \mathcal{F}_{\mathop{\mathrm{colim}}\nolimits } : U \longmapsto \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} \mathcal{F}_ i(U) \]

There are clearly projection maps $\mathcal{F}_{\mathop{\mathrm{lim}}\nolimits } \to \mathcal{F}_ i$ and canonical maps $\mathcal{F}_ i \to \mathcal{F}_{\mathop{\mathrm{colim}}\nolimits }$. These maps satisfy the requirements of the maps of a limit (reps. colimit) of Categories, Definition 4.14.1 (resp. Categories, Definition 4.14.2). Indeed, they clearly form a cone, resp. a cocone, over $\mathcal{F}$. Furthermore, if $(\mathcal{G}, q_ i : \mathcal{G} \to \mathcal{F}_ i)$ is another system (as in the definition of a limit), then we get for every $U$ a system of maps $\mathcal{G}(U) \to \mathcal{F}_ i(U)$ with suitable functoriality requirements. And thus a unique map $\mathcal{G}(U) \to \mathcal{F}_{\mathop{\mathrm{lim}}\nolimits }(U)$. It is easy to verify these are compatible as we vary $U$ and arise from the desired map $\mathcal{G} \to \mathcal{F}_{\mathop{\mathrm{lim}}\nolimits }$. A similar argument works in the case of the colimit.


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