## 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

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

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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)