Remark 4.14.4. We often write $\mathop{\mathrm{lim}}\nolimits _ i M_ i$, $\mathop{\mathrm{colim}}\nolimits _ i M_ i$, $\mathop{\mathrm{lim}}\nolimits _{i\in \mathcal{I}} M_ i$, or $\mathop{\mathrm{colim}}\nolimits _{i\in \mathcal{I}} M_ i$ instead of the versions indexed by $\mathcal{I}$. Using this notation, and using the description of limits and colimits of sets in Section 4.15 below, we can say the following. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram.

1. The object $\mathop{\mathrm{lim}}\nolimits _ i M_ i$ if it exists satisfies the following property

$\mathop{Mor}\nolimits _\mathcal {C}(W, \mathop{\mathrm{lim}}\nolimits _ i M_ i) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{Mor}\nolimits _\mathcal {C}(W, M_ i)$

where the limit on the right takes place in the category of sets.

2. The object $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ if it exists satisfies the following property

$\mathop{Mor}\nolimits _\mathcal {C}(\mathop{\mathrm{colim}}\nolimits _ i M_ i, W) = \mathop{\mathrm{lim}}\nolimits _{i\in \mathcal{I}^\text {opp}} \mathop{Mor}\nolimits _\mathcal {C}(M_ i, W)$

where on the right we have the limit over the opposite category with value in the category of sets.

By the Yoneda lemma (and its dual) this formula completely determines the limit, respectively the colimit.

There are also:

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

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).