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added the statement of the Fubini theorem for ends to a new section Properties.
(I wish this page would eventually give a good introduction to ends. I remember the long time when I banged my head against Kelly’s book and just didn’t get it. Then suddenly it all became obvious. It’s some weird effect with this enriched category theory that some of it is obvious once you understand it, but looks deeply mystifying to the newcomer. Kelly’s book for instance is a magnificently elegant resource for everyone who already understands the material, but hardly serves as an exposition of the ideas involved. I am hoping that eventually the nLab entries on enriched category theory can fill this gap. Currently they do not really. But I don’t have time for it either.)
There are some things where one can read and understand each single definition, and still not get the hang of how the subject ticks.
We’d need somebody familiar with category theory but unfamiliar with enriched category theory to tell us about his feelings when reading our entries on enriched category theory.
(The page on extranatural transformations is indeed nice, by the way. )
Ends and coends made a lot more sense to me when I started thinking of them as tensor products and homs of (pro)functors. The definition as a universal extranatural transformation never really clicked for me until after that. Specifically, I have found that it sometimes helps people not to think about the definition in terms of an arbitrary functor of two variables, but rather just as a tensor product or hom between two functors of one variable.
One other thing that it might be helpful to add to the page would be an leisurely explanation of why the end of a homset $\int_c hom(F c,G c)$ is the set of natural transformations from $F$ to $G$. This would then help motivate why enriched ends are used to define enriched objects of natural transformations.
Interesting. I had a similar experience:
I had been looking at Kelly’s book with the impression that I see the columns of green letters, but not the Matrix encoded by them, if you know what I mean.
Then at some point I drew a big diagram expressing the descent omega-category of an omega-category valued presheaf on some cover, which seemed to me to be a simplification of the expression in Street’s article. I understood that, but was wondering about that big diagram. Then Dominic Verity in a blog comment dropped the remark that this diagram expresses an end. After he said that, I could suddenly see all of the Matrix behind the green letters.
And now with hindsight, it is so obvious that I feel a bit upset with myself that I had that trouble at the beginning. It was this experience that made me think that if a book like Kelly’s had existed, but one which had taken the time to chat a bit about basic examples and waste a little time here and there on becoming friends with the concepts, I would have avoided wasting a certain amount of my life time on not getting the obvious.
But then I look now at the nLab material on enriched category theory that I coauthored, and I realize that I am just repeating history: the concept obvious to me I express in a few symbols in an nLab entry. Since it’s all obvious, it’s hard not to. Right? That’s the big problem with math in general. All ideas in math are obvious and evident. The trouble is communicating them via a bunch of black marks on paper.
Yeah. (-:
I still don’t get the obvious, but have hope :)
I added a reference to the $n$-Café discussion on Ends.
Is there a quick way to directly show, for $\mathcal{C}$ an enriched category, that $\mathcal{C}(c,-)$ preserves enriched ends, without passing through the general concept of weighted limits?
Well, take whatever definition of “enriched end” you are using, prove that it’s equivalent to the definition in terms of weighted limits, transfer the proof that $C(c,-)$ preserves weighted limits across that equivalence, and beta-reduce. (-:
My question was if there is a direct way, shortcutting this route, without invoking the general concept of weighted limits.
I would render Mike’s suggestion like this: for $F: J^{op} \otimes J \to C$, the end of $F$ if it exists is the representing object $\int_j F(j, j)$ for the functor $C^{op} \to V$ that takes $c$ to $V^{J^{op} \otimes J}(J(-, -), C(c, F(-, -)))$. So we have $V$-natural isomorphisms
$\array{ C(c, \int_j F(j, j)) & \cong & V^{J^{op} \otimes J}(J(-, -), C(c, F(-, -))) & \\ & \cong & \int_{j, j'} J(j, j') \multimap C(c, F(j, j')) & end\; formula\; for\; functor\; category\; hom \\ & \cong & \int_j \int_{j'} J(j, j') \multimap C(c, F(j, j')) & Fubini\; formula \\ & \cong & \int_j C(c, F(j, j)) & Yoneda\; lemma }$My suggestion was a way to derive a direct way, by starting out with the general concept of weighted limits and then eliminating it, as Todd has done.
Todd, thanks, that is the derivation via the end realized as a weighted limit. I was hoping for something that would not need to introduce this concept. But I guess in the end it will be easiest to just introduce it after all.
You don’t have to introduce the concept of “weighted limit” if you don’t want – the exact text of #12 never mentions it.
Alternatively, consider that in $Set$-enriched category theory, limits in categories are ultimately reducible to limits in $Set$, being jointly preserved and reflected by representables. Similarly, once the notion of an end in a base of enrichment $V$ has been established, say following Kelly’s book, then the only sensible notion of end $\int_j F(j, j)$ in a general $V$-category $C$ is simply dictated by the requirement that $C(-, \int_j F(j, j))$ takes $c \in C$ to the appropriate end in $V$, i.e., by the requirement that the representables $C(c, -)$ jointly preserve and reflect that end. In other words, arrange matters so that representables preserve ends by the very definition of end. The only work is a consistency check that this applies to $C = V$, but this is trivial.
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