# Why are (co-)ends called “(co-)ends”?

Briefly put, the (Co-)end is the universal wedge of a diagram.

Why is it called (co-)end? What is it an "end" of? By the universal property, it is in some sense the "universal end/finish" of some diagram but the same can be said about (co-)limits, so that doesn't seem too convincing to me.

• The earliest use of "(co)end" being used to describe this structure (that I can find) is Day–Kelly's Enriched functor categories. However, they don't give any justification for their terminology. – varkor Sep 15 '20 at 12:50
• @varkor Interesting, thanks! According to arxiv.org/abs/1501.02503 (p. v-vi), they were introduced even earlier, namely in Yoneda's paper "Ext and exact sequences" in 1960. – Qi Zhu Sep 15 '20 at 12:54
• I would venture that ends, being similar but distinct from limits, are called so because "end" and "limit" have similar but distinct meanings. – Zhen Lin Sep 15 '20 at 13:34

## 1 Answer

I have no proof this is the real reason, but I came to believe the name "end" arose to give a nod at "limit": very old notes (before Mac Lane there weren't real textbooks around..) call "root" what we today call "limit".

For limits, the modern metaphor became a standard because it's more elegant and suggestive (I mean: root? Seriously?), and because "limits, when they exist, are unique" becomes a true statement in a category as well as in a Hausdorff space.

But! "Ends" have been a thing in general topology since Freudhental: https://en.wikipedia.org/wiki/End_(topology)

Given a topological space you can cook a functor $${\bf Top}_* \to\bf Set$$ sending a space $$X$$ to its set of "ends", i.e. limits (in the categorical sense) of op-chains of $$\pi_0$$'s of complements of compact subsets of $$X$$ (whew, that was a nested definition).

Of course this begs a lot of questions (about the properties of that functor: coends have something to do with pi_0's, in the sense that the coend of $$\hom_C$$ is the $$\pi_0$$ of a certain arrow category of endomorphisms in $$C$$)! But most of all: how comes Freudhental defined something that is a functor (and not an easy one...), without having the word "functor" in mind?

• Great to see an answer from you, thank you! So you're saying that because ends are similar to limits and there is even a topological notion of ends that is constructed via limits - people may have called it "end" inspired by these observations? About the last question: Isn't it rather simply because many (/most) nice constructions are functorial - and the same just turns out to be the case for topological ends? – Qi Zhu Sep 16 '20 at 11:42