9
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$ – varkor Sep 15 '20 at 12:50
  • $\begingroup$ @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. $\endgroup$ – Qi Zhu Sep 15 '20 at 12:54
  • 4
    $\begingroup$ I would venture that ends, being similar but distinct from limits, are called so because "end" and "limit" have similar but distinct meanings. $\endgroup$ – Zhen Lin Sep 15 '20 at 13:34
5
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ 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? $\endgroup$ – Qi Zhu Sep 16 '20 at 11:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.