# What is the origin of the expression “Yoneda Lemma”?

Thank you very much in advance for telling where the expression “Yoneda Lemma” comes from.

EDIT 1. On page -14 of

Reprints in Theory and Applications of Categories, No. 3, 2003. Abelian Categories, by Peter J. Freyd

http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html

The Yoneda lemma turns out not to be in Yoneda’s paper. When, some time after both printings of the book appeared, this was brought to my (much chagrined) attention, I brought it the attention of the person who had told me that it was the Yoneda lemma. He consulted his notes and discovered that it appeared in a lecture that MacLane gave on Yoneda’s treatment of the higher Ext functors. The name “Yoneda lemma” was not doomed to be replaced.

EDIT 2. In the entry “Grothendieck functor” of the Encyclopaedia of Mathematics [EOM], edited by Michiel Hazewinkel”, one reads

In the English literature, the Grothendieck functor is commonly called the Yoneda embedding or the Yoneda–Grothendieck embedding.

EDIT 3. The article

Grothendieck, Alexander. Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules. Séminaire Bourbaki, 5 (1958-1960). Exposé No. 195, 22 p. Février 1960.

quoted in the EOM entry mentioned is available here.

EDIT 4. Subquestion 1: When was the Yoneda Lemma stated in print for the first time? Subquestion 2: When did this Gare du Nord conversation mentioned by Theo Buehler occur?

The advantage of Subquestion 1 is that it’s more likely to have a definite answer. [I think we all agree on the fact that Grothendieck’s Exposé does contain the “Yoneda Lemma”.]

EDIT 5. Tentative answer to Subquestion 1: I feel (tell me if I’m wrong) there is a consensus on the opinion that the Yoneda Lemma was stated in print for the first time in

Grothendieck, Alexander. Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules. Séminaire Bourbaki, 5 (1958-1960). Exposé No. 195, 22 p. Février 1960. Available here.

[I’m taking this opportunity to thank Theo Buehler for his generous contribution to this thread in particular, and to this site in general.]

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I thought it's a lemma that was proven by one called Yoneda :) –  Alexei Averchenko Jul 25 '11 at 13:29
@Alexei Averchenko - I’d like to know where precisely it was proved, and in which form. –  Pierre-Yves Gaillard Jul 25 '11 at 13:48
Well, that is not really the same question, right? That is the question "What is the origin of the Yoneda lemma?" –  Qiaochu Yuan Jul 25 '11 at 14:21
@Qiaochu Yuan - I’m not getting your point. The question is “What is the origin of the expression “Yoneda Lemma”?”. Of course, if you want to answer the question “What is the origin of the Yoneda Lemma?”, you’re more than welcome. (What do you exactly mean by “What is the origin of the Yoneda Lemma?”?) –  Pierre-Yves Gaillard Jul 25 '11 at 15:00
@Alexei Averchenko - How do you know that he proved it? –  Pierre-Yves Gaillard Jul 25 '11 at 15:28

Disclaimer: I'm posting this answer to back up Zhen Lin's answer to the end that Mac Lane is responsible for that terminology. Unfortunately, I can't find a first hand quote by Mac Lane online, but when I asked myself the same question several years ago, I tracked down many sources, but I can't find my notes on that at the moment, so I'm quoting from my unreliable memory and all has to be taken with a grain of salt. However, it is definitely worth getting copies of the references I provide below that aren't available online.

Edit 3: In a nutshell my answer boils down to the following: Organize a copy of issue 47 (1) (1998) of Mathematica Japonica and read the articles by Mac Lane (page 156) and Kinoshita (page 155). Snippet views of that issue are available on Google Books.

Here's a somewhat more extensive quote from the "Notes" on pages 77f of Mac Lane's Categories for the Working Mathematician:

The Yoneda Lemma made an early appearance in the work of the Japanese pioneer N. Yoneda (private communication to Mac Lane) [1954]; with time, its importance has grown.

Representable functors probably first appeared in topology in the form of "universal examples", such as the universal examples of cohomology operations (for instance, in J. P. Serre's 1953 calculations of the cohomology, modulo 2, of Eilenberg- Mac Lane spaces).

I think that in the first paragraph there is simply a comma missing between the parentheses and the square brackets referring to the 1954 article by N. Yoneda, On the homology theory of modules, J. Fac. Sci. Tokyo, Sec. I. 7, 193–227 (1954). MathSciNet review by H. Cartan: MR 68832 (in French). Edit: Of course, this could also be interpreted to mean that the meeting between Mac Lane and Yoneda at the Gare du Nord (see below) took place in 1954.

If I remember correctly, Yoneda doesn't prove his lemma in that article (see also the quote by Freyd in Edit 1 of the question). However, he proves that the left derived functors of a (right exact) functor $F: {}_R\mathbf{Mod} \to \mathbf{Ab}$ can be computed as $L_{q}F(A) = [\operatorname{Ext}_{R}^{q}((A,{-}),F]$ where the right hand side denotes the abelian group of natural transformations from the $q$th Yoneda Ext to $F$. In the course of the proof he essentially establishes the Yoneda Lemma for $R$-modules (which is the case $q = 0$ of course). See also Yoneda's follow-up paper On $\operatorname{Ext}$ and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 507–576. MathScinet review by G. S. Rinehart: MR 226854.

Edit 4: It may be off-topic but I think it's still worth pointing out, as it isn't as well-known as it deserves to be: Yoneda's second paper introduces what is nowadays called an exact category in the sense of Quillen, under the name of quasi-abelian $\mathscr{S}$-category. See the historical note on p.3f of my survey article for more on that (preliminary version available as arXiv:0811.1480 where the note is on p.4). Note that Yoneda's paper precedes Quillen's seminal Higher algebraic $K$-theory, I, Springer LNM 341 (1973), 85-147 by 13 years. MathSciNet review of Quillen's paper by S. M. Gersten: MR 338129.

Furthermore, there is the story that Yoneda and Mac Lane met in Paris at the Gare du Nord, where Mac Lane learned about it:

When he [Yoneda] arrived in Princeton, Eilenberg had moved (sabbatical?) to France (or maybe, Eilenberg left US just after Yoneda's arrival). So, Yoneda went to France a year later. At that time, Saunders Mac Lane was visiting category theorists, apparently to obtain information to write his book (or former survey), and he met the young Yoneda, among others. The interview started in a Café at Gare du Nord, and went on and on, and was continued even in Yoneda's train until its departure. The contents of this talk was later named by Mac Lane as Yoneda lemma. So, the famous Yoneda lemma was born in Gare du Nord. This must have been a good memory for Yoneda; I heard him tell this story many times. I do not know whether Mac Lane managed to leave the train before departure!

Edit 2: This excerpt is from an email by Yoshiki Kinoshita on occasion of Yoneda's death. From what I could see on Google Books the above paragraph appeared in polished form on page 155 in issue 47 (1) (1998) of Mathematica Japonica. See also Kinoshita's article A bicategorical analysis of $E$-categories, Mathematica Japonica, 47(1), 157-169, 1998.

See also the first paragraph on p.3 of C. McLarty's article Saunders Mac Lane and the Universal in Mathematics, Scientiae mathematicae Japonicae 19 (2006) 25–28:

Rather than recount the often told collaboration with Eilenberg, let us focus on the most famous lemma in category theory. Many aspects of Mac Lane’s thought are captured in the history and the mathematics of this result. Mac Lane was passionate about organizing and building the knowledge of category theory. He knew Nobuo Yoneda’s work in homology and so when they met in Paris Mac Lane eagerly talked with him about his wider, unpublished perspective on the methods. Mac Lane’s care as a historian of mathematics shows in his account of learning this lemma from Yoneda on a platform of the Gare du Nord waiting for Yoneda’s train (Mac Lane 1998b).

Reference 1998b in McLarty is: Mac Lane, The Yoneda lemma, Mathematica Japonica 47, 156, which unfortunately I could not locate online.

Finally, there is a passage in Mac Lane's autobiography telling the story on the Gare du Nord and I distinctly remember that Buchsbaum said that he learned about the Yoneda Lemma from Mac Lane in lectures on category theory.

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To address the question when the meeting at the Gare du Nord happened, it must have been some time between 1952-1956, but I can't find a reference stating exactly when Y. was in Paris. –  t.b. Jul 25 '11 at 17:05
Great answer. I'm willingly accepting your point that Yoneda first discovered "his" functor and lemma. Nevertheless, I find surprising that Grothendieck didn't mention it. I mean, it doesn't seem that Grothendieck needed to "steal" nobody's results, does it? –  a.r. Jul 25 '11 at 17:34
@Agustí: Thanks! Given that Grothendieck was so quick at internalizing and grasping any kind of categorical thought, I think he may well have discovered it and its importance on his own. Lacking an adequate reference may well be a sufficient explanation for not citing anything. I don't think I ever read anything from him claiming credit for the Yoneda lemma. –  t.b. Jul 25 '11 at 17:44
+1, Mac Lane used to tell this story on various occasions. –  Michal R. Przybylek Jul 25 '11 at 18:01
I agree with Agusti, in particular with the fact that your answer is a great one (thank you!). I believe (and I agree it’s an irrational belief) that if Grothendieck had thought that, say, Proposition 1.1 of his paper quoted above was not due to him, he would have mentioned it explicitly (even in the absence of an "adequate reference"). Grothendieck stated this "lemma" in many of his texts, and I find striking (if true) the fact that he never said it was due to somebody else. –  Pierre-Yves Gaillard Jul 25 '11 at 18:16

According to Mac Lane [CWM, 1998, p.77]:

The Yoneda Lemma made an early appearance in the work of the Japanese pioneer N. Yoneda (private communication to Mac Lane) [1954]; with time, its importance has grown.

N. Yoneda refers to Nobuo Yoneda (米田信夫).

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Thank you very much for your answer. Could you be more specific and describe as precisely as possible which statement of Yoneda Mac Lane is referring to. If you could give a reference to a paper of Yoneda, it’d be even nicer. –  Pierre-Yves Gaillard Jul 25 '11 at 13:40
@Pierre: The bibliography in CWM lists a 1954 paper of Yoneda: ‘On the homology theory of modules’, J. Fac. Sci. Tokyo, Sec. I. 7, 193–227 (1954). But Mac Lane's quote says ‘private communication’, so I don't think is what he's referring to. –  Zhen Lin Jul 25 '11 at 13:47
I find Mac Lane’s phrasing strange: he says it’s a private communication, but quote a paper. Have you looked at this paper? If Yoneda published his lemma, where was it? If he only talked about it, with whom? With Mac Lane, and nobody else? (I agree that all this is possible.) –  Pierre-Yves Gaillard Jul 25 '11 at 15:16
@Pierre-Yves: You won't find the Yoneda lemma in this form in the 1954 article by Yoneda. There are several places in which Mac Lane states that he met Yoneda in the Gare du Nord and he learned about the lemma there (if you can track down this issue of Mathematica Japonica Vol. 47, No. 1, you'll find it there or in Mac Lane's autobiography). See here par.4 for an online version of that story. I think I also remember Buchsbaum stating that he learned the statement from lectures by Mac Lane. –  t.b. Jul 25 '11 at 16:06
@Theo Buehler - Thanks a lot! I’ll look at all this. I definitely think you should turn your comments into an answer! –  Pierre-Yves Gaillard Jul 25 '11 at 16:16