The Stacks project

Lemma 4.14.10. Let $\mathcal{I}$, $\mathcal{J}$ be index categories. Let $M : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ be a functor. We have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ j M_{i, j} = \mathop{\mathrm{colim}}\nolimits _{i, j} M_{i, j} = \mathop{\mathrm{colim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i M_{i, j} \]

provided all the indicated colimits exist. Similar for limits.

Proof. Omitted. $\square$

Comments (2)

Comment #8886 by Laurent Moret-Bailly on

More precisely, if one of the double colimits exists then it is the total colimit.

Comment #8986 by Laurent Moret-Bailly on

Even more precisely: assume that exists for all . Then the resulting functor has a colimit if and only if does, and then the colimits coincide. (And if this is the case, it does not follow that the colimits with fixed exist).

There are also:

  • 7 comment(s) on Section 4.14: Limits and colimits

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Your email address will not be published.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 002M. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 4.14.10 as pdf