The Stacks project

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 (resp. 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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