The Stacks project

Definition 4.14.1. A limit of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ in $\mathcal{C}$ together with morphisms $p_ i : \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \to M_ i$ such that

  1. for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $p_{i'} = M(\phi ) \circ p_ i$, and

  2. for any object $W$ in $\mathcal{C}$ and any family of morphisms $q_ i : W \to M_ i$ (indexed by $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $q_{i'} = M(\phi ) \circ q_ i$ there exists a unique morphism $q : W \to \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ such that $q_ i = p_ i \circ q$ for every object $i$ of $\mathcal{I}$.

Comments (0)

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Your email address will not be published.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