The Stacks project

Remark 7.20.5. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. Given morphisms $g : u(U) \to V$ and $f : W \to V$ in $\mathcal{D}$ we can consider the functor

\[ \mathcal{C}^{opp} \longrightarrow \textit{Sets},\quad T \longmapsto \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(T, U) \times _{\mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(T), V)} \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(T), W) \]

If this functor is representable, denote $U \times _{g, V, f} W$ the corresponding object of $\mathcal{C}$. Assume that $\mathcal{C}$ and $\mathcal{D}$ are sites. Consider the property $P$: for every covering $\{ f_ j : V_ j \to V\} $ of $\mathcal{D}$ and any morphism $g : u(U) \to V$ we have

  1. $U \times _{g, V, f_ i} V_ i$ exists for all $i$, and

  2. $\{ U \times _{g, V, f_ i} V_ i \to U\} $ is a covering of $\mathcal{C}$.

Please note the similarity with the definition of continuous functors. If $u$ has $P$ then $u$ is cocontinuous (details omitted). Many of the cocontinuous functors we will encounter satisfy $P$.

Comments (0)

There are also:

  • 5 comment(s) on Section 7.20: Cocontinuous functors

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09W7. Beware of the difference between the letter 'O' and the digit '0'.