Talk:Yoneda lemma

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


 * [[Image:YonedaLemma-01.png]]

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 (talk • contribs) 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 (talk • contribs) 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)

The content and the complexity of this article
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 (talk • contribs) 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)


 * One can know what the Yoneda lemma is, and still read this article and find it daunting. Something about the way its written, the way it explains things. The Yoneda lemma is not that hard, in the end, but what's written here makes it sound overwhelmingly complicated. I can't quite put my finger on what the core problem is, so can't actually fix it. 67.198.37.16 (talk) 01:29, 8 March 2018 (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)


 * I agree w/ removal of the tag; however, let me repeat a comment above (viz it really is too technical, in a certain sense): One can know what the Yoneda lemma is, and still read this article and find it daunting. Something about the way its written, the way it explains things. The Yoneda lemma is not that hard, in the end, but what's written here makes it sound overwhelmingly complicated. I can't quite put my finger on what the core problem is, so can't actually fix it.  Perhaps examples taken from common (pre-)sheaves would help?? 67.198.37.16 (talk) 01:32, 8 March 2018 (UTC)

This article absolutely is too technical, so I am readding the "technical" tag. Please remember Wikipedia is an encyclopedia, not a maths textbook. A reader with a degree in theoretical physics and some familiarity with group theory would not understand even the introduction. I have just read Wikipedia's guidelines on technical articles and would recommend anyone else working on this article do the same. I think work is needed on simplifying this article - the introduction should at least be understandable! Let me elaborate a bit.

2nd sentence:

"It is an abstract result on functors of the type morphisms into a fixed object."

What does "morphisms into a fixed object" mean? There is no explanation nor any link to another Wikipedia article explaining what this means.

3rd sentence:

"It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms)."

One would think that putting the part in parentheses: "(viewing a group as a miniature category with just one object and only isomorphisms)" would be providing helpful explanation, but it just makes things more confusing for someone without the necessary background.

4th sentence:

"It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category."

One would again think that putting the part in parentheses: "(contravariant set-valued functors)" would be some helpful additional info. Instead it is just makes things even more confusing. There is no explanation of what "contravariant set-valued functors" are, nor a link to another Wikipedia article.

I would suggest the introduction be reworked like this:

"In mathematics, the Yoneda lemma is arguably the most important result in category theory.[1] It is a vast generalisation of Cayley's theorem from group theory and an important tool underlying several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda."

ideally followed by some explanation in very simple terms, with an example if possible. If this topic really is just too technical that no adequate simplification is possible, then I think the introduction should say so this in order to warn readers.

2A00:23C7:9386:BA00:BFE5:C6FD:704:D813 (talk) 16:07, 9 August 2021 (UTC)

Definition of notation used in the formal statement
I think the notation used in the equation
 * $$h^A \colon f \longmapsto \mathrm{Hom}(A,f) = [\![ \mathrm{Hom}(A,X) \ni g \mapsto f \circ g \in \mathrm{Hom}(A,Y) ]\!] $$

is uncommon. While I'm able to piece together its meaning from the surrounding paragraph, this kind of special notation needs a definition or reference. siddharthist (talk) 05:07, 3 October 2017 (UTC)
 * I just now broke this up into smaller bite-size pieces that should make it a little more readable. However, this entire article remains opaque. 67.198.37.16 (talk) 01:26, 8 March 2018 (UTC)

Links into the void
It is tremendously unfortunate not only that the connection between Cayley's theorem and Yoneda's lemma is left unexplained, but I also have to click on Cayley's theorem to find out which theorem is being called Cayley's theorem in this article.

On the other hand, I wish the writer had recognized that the important bit of information is not the name "Cayley" but the content of the theorem. In that case perhaps another 30 seconds of typing could have included a phrase like "the theorem that every finite group is a subgroup of the symmetric group on its elements" and thereby saved every future reader of this article multiple clicks into the void.

Why do I say "into the void"? Because in a large fraction of Wikipedia math articles, it is entirely unclear just how the linked Wikipedia article or section is related to the point that the editor was trying to make. Then one has to spend time hunting for the relevant passage, a hunt that often ends in failure.

And when you realize that every future visitor to the article will be tempted to click into the same void again ... it would be much better if just a few more words were added to the original article instead of a link into the void.2600:1700:E1C0:F340:5548:746F:48A8:5BCC (talk) 22:46, 23 October 2018 (UTC)


 * I wholeheartedly agree but don't have the qualifications to make the edit 80.99.107.132 (talk) 03:49, 17 April 2020 (UTC)A concerned citizen

"Yoneda lambda" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Yoneda lambda and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 September 23 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 08:27, 23 September 2022 (UTC)