# Talk:Yoneda lemma

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Field: Foundations, logic, and set theory

## Commutative diagram

Is there any was of cleaning up the commutative diagram...unfortunately the AMS CD package doesn't work. One could convert to JPEG I suppose (YUCK)

PLEASE o please could we get rid of "Philosophy" as a title head? When somebody says "My philosophy of the matter is.." I usually expect complete garbage to follow. This is not an attack on Philosophy, quite the contrary, it's an attack on the continual devaluation of Philosophy as a subject of intellectual endeavor. To use it as synonym for vagueness is unfortunate. User:CSTAR

Well, philosophy is no more. It's quite an old article by WP standards, and probably needs work. More about representable functors, eg in homotopy theory, would be handy also.

Charles Matthews 20:17, 5 May 2004 (UTC)

I replaced the original ASCII diagram

```    D --> Fun(Dop,Set)
|          |
|          |
|          |
V          V
C --> Fun(Cop,Set)
```

with the picture

The original seemed backwards to me the way things were stated. Please let me know if I screwed it up (highly possible, as things like covariant functors into categories of contravariant functors really make my head hurt). -- Fropuff 16:55, 2004 Jul 20 (UTC)

## Comment

Someone should probably say in what way the Yoneda lemma is a "vast generalisation of Cayley's theorem from group theory". Also, might be worth including the enriched-category version of the lemma as well. (hinted at at the bottom, but not stated explicitly) —Preceding unsigned comment added by 134.226.81.3 (talkcontribs) 02:25, 20 January 2006

Indeed. Feel free to make the edits yourself if you are so inclined. We always need more contributors -- Fropuff 05:02, 20 January 2006 (UTC)
It amounts to the same, but one can rephrase it as As a special case, when the category has only one object, and its morphisms correspond to the elements of a group, one recovers Cayley's theorem on realising a group as a permutation group.Hillgentleman 12:15, 11 September 2006 (UTC)
I think there should be a hyperlink to the article Nobuo Yoneda — Preceding unsigned comment added by 218.217.60.211 (talkcontribs) 13:41, 8 April 2007 (UTC)

## Natural functor?

Does the term "natural functor" have any technical significance? (That phrase appears in the description of hom-functors.) If so, I would appreciate it if somebody clarified it (or provided a link). If not, it would probably be better to change the language, since newbies (like me) might be confused from the tendency of "natural" to have a technical meaning in category theory. Thanks, 156.56.153.77 (talk) 04:44, 19 November 2009 (UTC)

## Proof

I think a bit more rigor is needed for the proof. What happens if C(A,x) is empty? How can we talk about an arrow then? I think the proof should like this: We need to show for each object x of C, the arrow Tau(x) is characterized by u=Tau(A)(idA). The set C(A,x) is either empty or it's not. If it's not empty we proceed as in the article. If it was empty, then the arrow Tau(x) must be the empty function from C(A,x) to F(x), and there's only one such empty function. Hence Tau is characterized by u. Empty hom sets occur a lot so this is not just nit picking Money is tight (talk) 04:30, 18 May 2010 (UTC)

That doesn't seem to be an issue. The empty set is an initial object in Set. 166.137.141.197 (talk) 11:53, 24 May 2010 (UTC)
Right. That's what I meant when I said "and there's only one such empty function". I just think this should be mentioned in the article and I'm not very good with latex. Money is tight (talk) 13:09, 24 May 2010 (UTC)

## fully faithful implies embedding?

In the section about the embedding it is said

..."the functor h is fully faithful, and therefore gives an embedding of Cop in the category of functors to Set."

I don't see this implication. A fully faithful functor and an embedding are two things, aren't they? It is not hard to prove that the Yoneda functor actually is an embedding, but in my opinion it does neither follow from the lemma nor from the functor being fully faithful. Please comment on this. Quiet photon (talk) 06:56, 12 October 2010 (UTC)

I believe this article should be 100% rewritten. First, it is formulated in terms of Set category, which is misleading and narrows the discourse. Second, it contains tons of very vaguely related information.

I believe I could rewrite it so that it would take half a page. But I am not sure what are the mores of this segment of wikipedia; can I? —Preceding unsigned comment added by Vlad Patryshev (talkcontribs) 09:13, 11 November 2010 (UTC)

It is better to proceed in incremental changes, meaning gradual improvement. Tkuvho (talk) 12:47, 2 February 2011 (UTC)
To help beginners, it may be helpful to elaborate a bit on the analogy with Cayley's theorem. Tkuvho (talk) 12:50, 2 February 2011 (UTC)

## Reference needed

It would be nice to have a reference to Yoneda's original publication. — Preceding unsigned comment added by 86.166.164.77 (talk) 12:39, 30 August 2013 (UTC)

According to Math StackExchange, the origin of the term is in a lecture Yoneda gave. APerson (talk!) 16:59, 21 December 2013 (UTC)

## Smallness

There is a problem with smallness. The Yoneda lemma doesn't need the category C to be small, but for the Yoneda embedding to be defined one does need it, as the category of functors C->Set is only really a category if C is small. Bruno321 (talk) 11:24, 4 February 2014 (UTC)

## Too technical?

Not for the intended readership, I think. I have removed the {{technical}} tag. Deltahedron (talk) 18:11, 23 February 2014 (UTC)