# Motivation for the integral notation $\int _{I \in \mathbb{I}} D(I)$ for a categorical limit?

What is the motivation for the integral notation $$\int _{I \in \mathbb{I}} D(I)$$ for a categorical limit, which is otherwise known as $$\underset {\mathbb{I} \leftarrow} {lim} \ D$$?

• I've only seen the integral notation for (co)ends. Where did you see it used for general categorical limits? – Olivier Bégassat Nov 9 '20 at 12:59
• @OlivierBégassat Hi. 30 secs into this introductory material here . – JRC Nov 9 '20 at 13:01
• Hm, indeed. The notation that is probably most common is $\mathrm{lim}$ for limits and $\mathrm{colim}$ for colimits. This way you don't have to remember which of $\int_{I\in\Bbb{I}}$ and $\int^{I\in\Bbb{I}}$, or $\underset{\Bbb{I}\leftarrow}\lim$ and $\underset{\Bbb{I}\rightarrow}\lim$ corresponds to what. As for your question proper, motivation, I'm afraid I don't know. One can speculate that in many contexts a colimit is a sort of sum, so the notation might make sense for colimits. – Olivier Bégassat Nov 9 '20 at 13:43
• Hmm. Interesting. Interesting possible rationale via co-limits. – JRC Nov 10 '20 at 9:36

As mentioned in the comments, this notation is most commonly used for ends and coends. In fact, limits and colimits are special cases of ends and coends (the case where one takes the end /coend of a functor $$F: C \times C^{op} \to D$$ which depends only on one of its variables), so this notation is a special case of the end / coend notation. For more discussion of this notation, see here.