From my home turf of neuroscience, the Hebbian learning law: neurons

Dear All, This thread prompted me to read: JEAN BENABOU (1932–2022): The man and the mathematician http://cahierstgdc.com/wp-content/uploads/2022/07/F.-BORCEUX-LXIII-3.pdf which made me think it would be nice to reprint seminal unpublished works of Professor Benabou (such as those discussed in the above) as: Reprints in Theory and Applications of Categories http://www.tac.mta.ca/tac/reprints/index.html simply because Professor Benabou's orginal conceptualization of many category theoretic constructs, beginning with closed to fibered categories, along with, of course, monads/triples (ibid., ref. 12), are worth studying in and of themselves and/or in the context of thier conceptual cousins, so to speak. Here are a couple of illustrations of conceptual kinship that is quite commonplace in science: Professor F. William Lawvere's Axiomatic Cohesion (http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf) & Professor Johnstone's Punctual Connectedness (http://www.tac.mta.ca/tac/volumes/25/3/25-03.pdf) Grothendieck: Descent (http://www.numdam.org/item/?id=SB_1958-1960__5__369_0) & Bastiani and Ehresmann: Sketches (http://www.numdam.org/item/CTGDC_1972__13_2_104_0.pdf) & F. William Lawvere: Functorial Semantics (http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf) that fire together wire together, which is credited to Donald Hebb (1949; https://drive.google.com/file/d/1_TCefN8KL36RXUA-12S3EZT57SKSoACq/view?usp=s..., p. 43) can be traced way back to William James (1890; ibid., p. 2). In closing, in response to my one too many emails on how sets (e.g., {a, b}) that are used to introduce set theory are not exactly Cantor's lauter Einsen (cf. {*, *}), Professor F. William Lawvere, while acknowledging it (spectrum vs. rank; https://conceptualmathematics.wordpress.com/2012/06/08/structure-of-internal...), helped me realize how history is not a home to stay put, but a resource to build on (https://conceptualmathematics.wordpress.com/2012/09/23/comfortable-with-sheh...). Here's one direction to move on i.e., build on Leibniz monad to get to intensive quality (e.g., idempotent; https://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf). Thanking you, Yours truly, posina On Thu, Nov 9, 2023 at 12:44 PM Urs Schreiber <urs.schreiber@googlemail.com> wrote:

Just to note that in 2009 on this same list, the lunch-genesis of "monad" was attributed to Jean Benabou, see

Barr 2009 https://ncatlab.org/nlab/show/monad#Barr09

Indeed, in print the term was introduced by

Benabou 1967: "Introduction to Bicategories" (section 5.4)

together with the astute observation that monads are the lax images of 1 and thus quite the 2-categorical version of the units=monads of Euclid.

https://ncatlab.org/nlab/show/monad+terminology

On Thu, Nov 9, 2023 at 1:22 AM Michael Barr, Prof. <barr.michael@mcgill.ca> wrote:

...

People seemed to enjoy my history of the founding of TAC, so I thought you might enjoy my sharing of other historical notes.

This construction was introduced in Godement's book Théorie des faisceaux in connection with his resolution of sheaves by "faisceaux mous" (soft sheaves) which are an injective class. He called this "la construction standarde". It is not clear whether this was intended to name them or merely describe them. At any rate, around 1960,

Benno Eckmann and his students took as a name and called them standard constructions. One of the students, Peter Huber, told me that they were having trouble, in particular cases, verifying the associative law. And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U. He wondered if every adjoint pair gave rise to a standard construction and proved that it did. Then another student, Heinrich Kleisli, showed that the converse was also true. That gave us the well-known Kleisli construction.

In 1963 Samuel Eilenberg and John Moore published a monograph called Foundations of Relative Homological Algebra in which they used this construction as basic. Only they didn't call them standard constructions; they called them triples. I once asked Sammy why and he replied that it didn't seem like an important concept and it didn't seem worth it to spend a lot of time worrying about the name. This is in stark contrast with the time he and Henri Cartan spent thinking about the name for their basic sequences. There is a story, perhaps apocryphal, that their book was in proof stage before they settled on the exact name.

So triple was name Jon Beck and I were using in our joint work on homological algebra. Then in 1966 there was a category meeting in Oberwohlfach and there was a lot of discussion of a better name. The next bit of the story comes out of my extremely fallible memory and could well be mistaken. One day at lunch I was sitting next to Anders Koch and he asked what I thought about the name monad. I thought (and still think!) it was a pretty good name and so he proposed it and the assembled crowd agreed and adopted it. I would have too, but Jon rejected it. Why, I asked him. He did not think it a good name and refused to use it. He said there was no point in replacing one bad name by another. Since we were collaborating and since he was even more stubborn than me, that's they way it was. In our papers, Jon insisted on putting functions to the right of their arguments, just like reverse Polish.

Then we stopped collaborating and, by 1980, I think I was about ready to start using monad. But then TTT came along and the alliteration was just too good to pass up. Charles Wells agreed on those grounds.

And what about fundamental construction? I spent six and a half months at the ETH in Zurich. A few days after I arrived, I got a phone call from Peter Huber, the aforementioned former student of Eckmann's. He had just received from Math Reviews a paper written by Jean-Marie Maranda that used that term for the concept and Huber asked me if there was any way to stop that proliferation of names. As far as I know, that was the only place that term was ever used.

Michael

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Dear Michael, Dear All Michael, many thanks for these historical notes. A small complement: It seems to me that, without the terminology, the ideas related to monads etc. are lurking behind S. Mac Lane [Homologie des anneaux et des modules, Louvain 1956]. I cannot get hold of a copy. I did not find it in TAC. Does anybody on this list have a copy? In his MR review, Buchsbaum writes "The definition of the construction is made over two abelian categories, thereby giving the standard constructions of homological algebra". Godement's terminology [1958] p. 270/71: - "ss object" ("objet semi-simplicial") for nowadays "cosimplicial object" - résolution simpliciale standard: a cosimplicial object which yields the resolution by soft sheaves - construction fondamentale: in the literature later termed monad Dold-Puppe [Homologie nicht-additiver Funktoren, 1961], in 9.20 p. 289 hint at the cosimplicial object behind the cobar construction. Rather than "cosimplicial object", they use the terminology "negative ss object" and put "co ss object" ("ko-s.s. Object" in German) in parentheses (9.2, 9.3 p.284). Truly minor: In ancient Greek (Euclid etc.), the terminology is monas for unit (not monad), with plural form monades. Leibniz introduced the (French) term "monade" in his book "La monadologie". In French, the correct wording is "construction standard", cf. above Godement's "résolution simpliciale standard". Best Johannes ________________________________ De: "Nathanael Arkor" <nathanael.arkor@gmail.com> À: categories@mq.edu.au Envoyé: Jeudi 9 Novembre 2023 07:03:42 Objet: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions Dear Michael, Thank you very much for sharing this piece of history. I, personally, am always deeply interested to learn of the stories behind the mathematics, and the people responsible for it, that have been fundamental in shaping what we understand to be category theory today. In the interest of posterity, I have a slight amendment to the etymology you described. I should preface what follows by making it clear that my understanding is based only on the literature, rather than personal experience, and so may not accurately reflect the actual history (in which case I would be glad to be corrected). (On fundamental/standard constructions) The terminology employed by Godement (on p. 271 of the 1958 "Topologie algébrique et théorie des faisceaux") was "la construction fondamentale", i.e. "the fundamental construction". It was thus Godement rather than Maranda who introduced this terminology; Maranda appears to be the only one who continued to use that terminology in later work. Huber then employed the terminology "standard construction" for the notion of comonad (in §2 of the 1961 "Homotopy Theory in General Categories") – monads are instead called "dual standard constructions". It does not appear to be until around 1968 that the terminology "standard construction" appears in reference to the concept of monad rather than comonad. (The seemingly misinformed assertion that Godement introduced the terminology "standard construction" appears, perhaps for the first time, in the 1969 Proceedings of "Seminar on Triples and Categorical Homology Theory", which may be where confusion has arisen.) (On triples) I believe the paper of Eilenberg and Moore in which the terminology "triple" first appears is the 1965 "Adjoint functors and triples" (I did not find the terminology "triple" in "Foundations of Relative Homological Algebra"). (Incidentally, in Dubuc's 1968 paper "Adjoint triangles", he refers to the terminology "triplex", but this appears to be a typo, as I cannot find that terminology elsewhere.) (On monads) In a 2009 email to the categories mailing list (https://www.mta.ca/~cat-dist/archive/2009/09-4), you recounted the same story about the origin of the terminology "monad", except that you recalled the one who proposed the terminology was Jean Bénabou. This seems likely, since, as far as I can tell, the term first appears in Bénabou's 1967 "Introduction to bicategories" (p. 39), where the terminology is justified in a footnote on p. 40. Best, Nathanael On 8 Nov 2023, 22:22 +0100, Michael Barr, Prof. <barr.michael@mcgill.ca>, wrote: People seemed to enjoy my history of the founding of TAC, so I thought you might enjoy my sharing of other historical notes. This construction was introduced in Godement's book Théorie des faisceaux in connection with his resolution of sheaves by "faisceaux mous" (soft sheaves) which are an injective class. He called this "la construction standarde". It is not clear whether this was intended to name them or merely describe them. At any rate, around 1960, Benno Eckmann and his students took as a name and called them standard constructions. One of the students, Peter Huber, told me that they were having trouble, in particular cases, verifying the associative law. And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U. He wondered if every adjoint pair gave rise to a standard construction and proved that it did. Then another student, Heinrich Kleisli, showed that the converse was also true. That gave us the well-known Kleisli construction. In 1963 Samuel Eilenberg and John Moore published a monograph called Foundations of Relative Homological Algebra in which they used this construction as basic. Only they didn't call them standard constructions; they called them triples. I once asked Sammy why and he replied that it didn't seem like an important concept and it didn't seem worth it to spend a lot of time worrying about the name. This is in stark contrast with the time he and Henri Cartan spent thinking about the name for their basic sequences. There is a story, perhaps apocryphal, that their book was in proof stage before they settled on the exact name. So triple was name Jon Beck and I were using in our joint work on homological algebra. Then in 1966 there was a category meeting in Oberwohlfach and there was a lot of discussion of a better name. The next bit of the story comes out of my extremely fallible memory and could well be mistaken. One day at lunch I was sitting next to Anders Koch and he asked what I thought about the name monad. I thought (and still think!) it was a pretty good name and so he proposed it and the assembled crowd agreed and adopted it. I would have too, but Jon rejected it. Why, I asked him. He did not think it a good name and refused to use it. He said there was no point in replacing one bad name by another. Since we were collaborating and since he was even more stubborn than me, that's they way it was. In our papers, Jon insisted on putting functions to the right of their arguments, just like reverse Polish. Then we stopped collaborating and, by 1980, I think I was about ready to start using monad. But then TTT came along and the alliteration was just too good to pass up. Charles Wells agreed on those grounds. And what about fundamental construction? I spent six and a half months at the ETH in Zurich. A few days after I arrived, I got a phone call from Peter Huber, the aforementioned former student of Eckmann's. He had just received from Math Reviews a paper written by Jean-Marie Maranda that used that term for the concept and Huber asked me if there was any way to stop that proliferation of names. As far as I know, that was the only place that term was ever used. Michael You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

[[Sent on behalf of ross.street@mq.edu.au]] Dear Michael Thank you for filling in all that history. Heinrich Kleisli sent his paper to Saunders Mac Lane as editor. Saunders told us that he advised Heinrich Kleisli that Eilenberg-Moore had solved the problem, as raised by Peter Hilton in his review of Huber's paper, on whether every monad was generated by an adjunction. I presume Heinrich was unaware of E-M at the time, and vive la difference! Another name that Saunders was testing, in his wonderful lectures at Bowdoin College (Maine) in the northern summer of 1969, was "triad". This was I think the second run through on the subject of his Graduate Text in Math #5 (I believe the first was at the Australian National University while I was in Illinois; and there was a third run at Tulane University where Eduardo Dubuc, Jack Duskin and I were after Bowdoin). The triad name did not survive. Incidentally, Bob Walters used the term "device" in his ANU thesis for the version that avoids the composite of the endofunctor with itself. When I mentioned that Jean Bénabou was the first to use the name "monad" in a publication (SLNM 47), Bill Lawvere said Sammy Eilenberg had come up with that name first. As Bill's student, Anders Kock may know more about that. In my dealings with Sammy, he never mentioned such a claim, but I had not asked him either. Speaking of bicategories, I know Jean visited Chicago and did ask Saunders permission to use "bicategory" since Saunders had used that term for a version of "factorization system". I do not know how much interaction Jean had with Sammy other than at category conferences. Ross ________________________________ From: Robert Pare <R.Pare@Dal.Ca> Sent: Friday, November 10, 2023 7:03 AM To: Michael Barr, Prof. <barr.michael@mcgill.ca>; Categories mailing list <categories@mq.edu.au> Subject: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions Mike, The usual caveat about memory applies. I came on the scene not long after this. My first big meeting was at the Battelle institute in Seattle in 1968. At that time Mac Lane was advocating forcefully for the name "triad" to replace "triple". Lambek was giving a series of talks on deductive systems and monads. His talk started "Let Trip be the category of standard constructions. A standard construction is a quadruple (A, T, eta, mu)..." Bob ________________________________ From: Michael Barr, Prof. <barr.michael@mcgill.ca> Sent: November 8, 2023 5:19 PM To: categories@mq.edu.au <categories@mq.edu.au> Subject: The game of the name: Standard constructions, triples, monads, fundamental constructions CAUTION: The Sender of this email is not from within Dalhousie. People seemed to enjoy my history of the founding of TAC, so I thought you might enjoy my sharing of other historical notes. This construction was introduced in Godement's book Théorie des faisceaux in connection with his resolution of sheaves by "faisceaux mous" (soft sheaves) which are an injective class. He called this "la construction standarde". It is not clear whether this was intended to name them or merely describe them. At any rate, around 1960, Benno Eckmann and his students took as a name and called them standard constructions. One of the students, Peter Huber, told me that they were having trouble, in particular cases, verifying the associative law. And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U. He wondered if every adjoint pair gave rise to a standard construction and proved that it did. Then another student, Heinrich Kleisli, showed that the converse was also true. That gave us the well-known Kleisli construction. In 1963 Samuel Eilenberg and John Moore published a monograph called Foundations of Relative Homological Algebra in which they used this construction as basic. Only they didn't call them standard constructions; they called them triples. I once asked Sammy why and he replied that it didn't seem like an important concept and it didn't seem worth it to spend a lot of time worrying about the name. This is in stark contrast with the time he and Henri Cartan spent thinking about the name for their basic sequences. There is a story, perhaps apocryphal, that their book was in proof stage before they settled on the exact name. So triple was name Jon Beck and I were using in our joint work on homological algebra. Then in 1966 there was a category meeting in Oberwohlfach and there was a lot of discussion of a better name. The next bit of the story comes out of my extremely fallible memory and could well be mistaken. One day at lunch I was sitting next to Anders Koch and he asked what I thought about the name monad. I thought (and still think!) it was a pretty good name and so he proposed it and the assembled crowd agreed and adopted it. I would have too, but Jon rejected it. Why, I asked him. He did not think it a good name and refused to use it. He said there was no point in replacing one bad name by another. Since we were collaborating and since he was even more stubborn than me, that's they way it was. In our papers, Jon insisted on putting functions to the right of their arguments, just like reverse Polish. Then we stopped collaborating and, by 1980, I think I was about ready to start using monad. But then TTT came along and the alliteration was just too good to pass up. Charles Wells agreed on those grounds. And what about fundamental construction? I spent six and a half months at the ETH in Zurich. A few days after I arrived, I got a phone call from Peter Huber, the aforementioned former student of Eckmann's. He had just received from Math Reviews a paper written by Jean-Marie Maranda that used that term for the concept and Huber asked me if there was any way to stop that proliferation of names. As far as I know, that was the only place that term was ever used. Michael You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. 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It's curious to observe what all but appears as reluctance to acknowledge Benabou's contribution. Even in adopting Benabou's terminology "monad", it seems to be generally overlooked that he offered that term with a substantial mathematical motivation beyond the somewhat shallow alliteration on "monoid": Monads are the lax images of the unit 1. It seems hard to argue that this was already folklore when the term was proposed, given that even now it is hardly ever mentioned. It looks like Benabou was ahead of his time. On Fri, Nov 10, 2023 at 12:37 AM JS Lemay <js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>> wrote: [[Sent on behalf of ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>]] Dear Michael Thank you for filling in all that history. Heinrich Kleisli sent his paper to Saunders Mac Lane as editor. Saunders told us that he advised Heinrich Kleisli that Eilenberg-Moore had solved the problem, as raised by Peter Hilton in his review of Huber's paper, on whether every monad was generated by an adjunction. I presume Heinrich was unaware of E-M at the time, and vive la difference! Another name that Saunders was testing, in his wonderful lectures at Bowdoin College (Maine) in the northern summer of 1969, was "triad". This was I think the second run through on the subject of his Graduate Text in Math #5 (I believe the first was at the Australian National University while I was in Illinois; and there was a third run at Tulane University where Eduardo Dubuc, Jack Duskin and I were after Bowdoin). The triad name did not survive. Incidentally, Bob Walters used the term "device" in his ANU thesis for the version that avoids the composite of the endofunctor with itself. When I mentioned that Jean Bénabou was the first to use the name "monad" in a publication (SLNM 47), Bill Lawvere said Sammy Eilenberg had come up with that name first. As Bill's student, Anders Kock may know more about that. In my dealings with Sammy, he never mentioned such a claim, but I had not asked him either. Speaking of bicategories, I know Jean visited Chicago and did ask Saunders permission to use "bicategory" since Saunders had used that term for a version of "factorization system". I do not know how much interaction Jean had with Sammy other than at category conferences. Ross ________________________________ From: Robert Pare <R.Pare@Dal.Ca> Sent: Friday, November 10, 2023 7:03 AM To: Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>>; Categories mailing list <categories@mq.edu.au<mailto:categories@mq.edu.au>> Subject: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions Mike, The usual caveat about memory applies. I came on the scene not long after this. My first big meeting was at the Battelle institute in Seattle in 1968. At that time Mac Lane was advocating forcefully for the name "triad" to replace "triple". Lambek was giving a series of talks on deductive systems and monads. His talk started "Let Trip be the category of standard constructions. A standard construction is a quadruple (A, T, eta, mu)..." Bob ________________________________ From: Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> Sent: November 8, 2023 5:19 PM To: categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto:categories@mq.edu.au>> Subject: The game of the name: Standard constructions, triples, monads, fundamental constructions CAUTION: The Sender of this email is not from within Dalhousie. People seemed to enjoy my history of the founding of TAC, so I thought you might enjoy my sharing of other historical notes. This construction was introduced in Godement's book Théorie des faisceaux in connection with his resolution of sheaves by "faisceaux mous" (soft sheaves) which are an injective class. He called this "la construction standarde". It is not clear whether this was intended to name them or merely describe them. At any rate, around 1960, Benno Eckmann and his students took as a name and called them standard constructions. One of the students, Peter Huber, told me that they were having trouble, in particular cases, verifying the associative law. And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U. He wondered if every adjoint pair gave rise to a standard construction and proved that it did. Then another student, Heinrich Kleisli, showed that the converse was also true. That gave us the well-known Kleisli construction. In 1963 Samuel Eilenberg and John Moore published a monograph called Foundations of Relative Homological Algebra in which they used this construction as basic. Only they didn't call them standard constructions; they called them triples. I once asked Sammy why and he replied that it didn't seem like an important concept and it didn't seem worth it to spend a lot of time worrying about the name. This is in stark contrast with the time he and Henri Cartan spent thinking about the name for their basic sequences. There is a story, perhaps apocryphal, that their book was in proof stage before they settled on the exact name. So triple was name Jon Beck and I were using in our joint work on homological algebra. Then in 1966 there was a category meeting in Oberwohlfach and there was a lot of discussion of a better name. The next bit of the story comes out of my extremely fallible memory and could well be mistaken. One day at lunch I was sitting next to Anders Koch and he asked what I thought about the name monad. I thought (and still think!) it was a pretty good name and so he proposed it and the assembled crowd agreed and adopted it. I would have too, but Jon rejected it. Why, I asked him. He did not think it a good name and refused to use it. He said there was no point in replacing one bad name by another. Since we were collaborating and since he was even more stubborn than me, that's they way it was. In our papers, Jon insisted on putting functions to the right of their arguments, just like reverse Polish. Then we stopped collaborating and, by 1980, I think I was about ready to start using monad. But then TTT came along and the alliteration was just too good to pass up. Charles Wells agreed on those grounds. And what about fundamental construction? I spent six and a half months at the ETH in Zurich. A few days after I arrived, I got a phone call from Peter Huber, the aforementioned former student of Eckmann's. He had just received from Math Reviews a paper written by Jean-Marie Maranda that used that term for the concept and Huber asked me if there was any way to stop that proliferation of names. As far as I know, that was the only place that term was ever used. Michael You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. 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Dear Urs, Jiri, and All It is not always clear from the publication record who thought of things first. Bénabou's Intro to bicategories was published by SLNM in 1967. This is before any Oberwolfach category conference. However, it is after the La Jolla 1965 conference where Eilenberg-Kelly presented closed categories with the type-set published paper appearing in 1966. By that time we had Mac Lane's coherence theorem for monoidal categories and Kelly's reduction of the axioms to two. Bénabou's multiplicative categories tried to incorporate coherence in the definition but there is a problem with his definition. In my opinion, Bénabou's contribution was to define bicategory (using the two axiom approach) as the several object version of monoidal category, thereby initiating weak higher category theory. This took courage. John Gray was already working with 2-categories, a concept of Charles Ehresmann. John told me in 1968-9 that he was convinced of the importance of bicategories because of the example of spans in a category with pullbacks. I was already convinced by Bénabou but felt the world might not be ready for papers using them. Also in the Eilenberg-Kelly paper, there were closed and monoidal functors. They agree in the closed monoidal case. These were of the lax rather than strong kind. I understand that Eilenberg pushed the presentation of their paper into emphasising closed over monoidal categories, but both are there. In Chapter IV Section 3 on examples, they point out that a closed functor from 1 to sets is a monoid and from I into abelian groups is a ring, etc.; they recognized that monoidal functors from 1 were monoids. Bénabou's morphisms of bicategories (lax functors) became the several object version. Jack Duskin told me in 1968 that Bénabou had the construction of a 2-category C' for each category C such that lax functors out of C amounted to 2-functors out of C'. I don't know how that fits historically with the result of Lawvere for C = 1, that André Joyal mentioned, which appeared in the ``Zurich triples book'' SLNM 80 (1969). Ross

On 12 Nov 2023, at 11:13 pm, Jirí Adámek <j.adamek@tu-bs.de> wrote:

Dear All,

Bill Lawvere once told me that 'monad' had been the idea of Eilenberg. Later I asked him by email about the details and he answered the following:

I do not remember in which year it was. (Maybe 1968, judging from vague allusions in Springer Lecture Notes 80.) In any case it was in the common room of the old castle at Oberwolfach when Sammy came out from behind the piano and announced the change. His informal speech emphasized that the word would inflect well: 'monadic' etc. He also explicitly said that nobody would ever confuse it with Leibnitzian monads.

Jiri

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Dear Mike Thank you so much for correcting my statement about Oberwolfach meetings. The 1966 meeting must have been the one Bill had meant. The meeting was presumably the first one enabled by Schubert and Gray. As you say, the most likely scenario is that Jean Bénabou mentioned monad to Sammy and Sammy bradcast it with his stamp of approval. Also, in my last paragraph below, it would have been the second half of 1969 (not 1968 when Jack Duskin told me about Bénabou's C \mapsto C'. It was part of the discussion at Tulane with Gray, Mac Lane and Duskin about whether Gray's closed structure on 2-Cat was monoidal. Ross On 14 Nov 2023, at 2:30 am, Michael Barr, Prof. <barr.michael@mcgill.ca> wrote: To set the record straight, there was definitely an Oberwohlfach meeting on CT in 1966. I met Bill and Fatima there as well Benabou, Kleisli, Kock, Laudal and others. Before that there was a small meeting in Chicago in April, 1965 and then the La Jolla meeting. I wasn't at the latter. I am more than willing to believe that it was Benabou sitting next to me who proposed monad. It is entirely possible that Sammy came down and pronounced it "official". And it was certainly in the old castle. Michael ________________________________ From: Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> Sent: Sunday, November 12, 2023 9:58 PM To: j.adamek <j.adamek@tu-bs.de<mailto:j.adamek@tu-bs.de>>; urs.schreiber <urs.schreiber@googlemail.com<mailto:urs.schreiber@googlemail.com>> Cc: JS Lemay <js.lemay@mq.edu.au<mailto:js.lemay@mq.edu.au>>; Categories mailing list <categories@mq.edu.au<mailto:categories@mq.edu.au>> Subject: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions Dear Urs, Jiri, and All It is not always clear from the publication record who thought of things first. Bénabou's Intro to bicategories was published by SLNM in 1967. This is before any Oberwolfach category conference. However, it is after the La Jolla 1965 conference where Eilenberg-Kelly presented closed categories with the type-set published paper appearing in 1966. By that time we had Mac Lane's coherence theorem for monoidal categories and Kelly's reduction of the axioms to two. Bénabou's multiplicative categories tried to incorporate coherence in the definition but there is a problem with his definition. In my opinion, Bénabou's contribution was to define bicategory (using the two axiom approach) as the several object version of monoidal category, thereby initiating weak higher category theory. This took courage. John Gray was already working with 2-categories, a concept of Charles Ehresmann. John told me in 1968-9 that he was convinced of the importance of bicategories because of the example of spans in a category with pullbacks. I was already convinced by Bénabou but felt the world might not be ready for papers using them. Also in the Eilenberg-Kelly paper, there were closed and monoidal functors. They agree in the closed monoidal case. These were of the lax rather than strong kind. I understand that Eilenberg pushed the presentation of their paper into emphasising closed over monoidal categories, but both are there. In Chapter IV Section 3 on examples, they point out that a closed functor from 1 to sets is a monoid and from I into abelian groups is a ring, etc.; they recognized that monoidal functors from 1 were monoids. Bénabou's morphisms of bicategories (lax functors) became the several object version. Jack Duskin told me in 1968 that Bénabou had the construction of a 2-category C' for each category C such that lax functors out of C amounted to 2-functors out of C'. I don't know how that fits historically with the result of Lawvere for C = 1, that André Joyal mentioned, which appeared in the ``Zurich triples book'' SLNM 80 (1969). Ross

On 12 Nov 2023, at 11:13 pm, Jirí Adámek <j.adamek@tu-bs.de<mailto:j.adamek@tu-bs.de>> wrote:

Dear All,

Bill Lawvere once told me that 'monad' had been the idea of Eilenberg. Later I asked him by email about the details and he answered the following:

I do not remember in which year it was. (Maybe 1968, judging from vague allusions in Springer Lecture Notes 80.) In any case it was in the common room of the old castle at Oberwolfach when Sammy came out from behind the piano and announced the change. His informal speech emphasized that the word would inflect well: 'monadic' etc. He also explicitly said that nobody would ever confuse it with Leibnitzian monads.

Jiri

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Dear John Well that is important for me to know/remember. It must have been **that** use of the term ``bicategory'' that Jean was seeking approval from Mac Lane to use for the several object form. I do think the terms closed and monoidal category are due to the Eilenberg-Kelly however there would have been discussion of terminology at the LaJolla conference. Very tricky! Ross

On 14 Nov 2023, at 9:14 am, John Baez <john.baez@ucr.edu> wrote:

Hi -

It is not very important, but I was amused to discover recently that Mac Lane's famous 1963 paper on monoidal categories, "Natural associativity and commutativity," does not mention "monoidal categories". Instead he called them "bicategories"!

Later in this paper he writes

"Bicategories have been introduced independently by several authors. They are in Bénabou [1], with a different but equivalent definition of "coherence," but without any finite list of conditions sufficient for the coherence."

This is not Bénabou's famous paper on bicategories: instead it's "Catégories avec multiplication", where Bénabou introduces a preliminary concept of monoidal category, which he called "catégorie avec multiplication".

Furthermore, it's now recognized that Bénabou's formulation of coherence for monoidal categories is not quite right. Benabou's version is along the lines of "all diagrams formed by associators and unitors commute", and he does not state this in a way that rules out problematic cases caused by coincidental equations between objects.

It seems the history of mathematics is endlessly tricky.

Best, John Baez

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...

It seems the history of mathematics is endlessly tricky.

we are watching in real time that all history is endlessly tricky and terrible. some of us settled on math precisely because it seemed less terribly tricky than the rest. i was under a solemn oath after the high-school that whatever i might do in my life the only requirement was that it should have nothing to do with math, which seemed like people competing who is smarter. i settled on it after i destroyed one relationship after another just proving stuff all the time. so it is a little ironic to find on the other end the history of mathematics being the tricky part. :) -- dusko

Ross Street <ross.street@mq.edu.au> writes: [snip]

however there would have been discussion of terminology at the LaJolla conference. Very tricky!

Ross

...

On 14 Nov 2023, at 9:14 am, John Baez <john.baez@ucr.edu> wrote:

Hi -

It is not very important, but I was amused to discover recently that [snip] out problematic cases caused by coincidental equations between objects.

It seems the history of mathematics is endlessly tricky.

Best, John Baez

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[[Sent on behalf of nathanael.arkor@gmail.com as it did get sent out/approved properly -- trying to fix this, apologies]] Dear all, While we are on the subject of provenance, and since it has been mentioned, it seems worth taking the opportunity to address the claim that Charles Ehresmann was responsible for the notion of 2-category. This assertion appears in Eilenberg and Kelly's 1965 La Jolla article "Closed Categories". However, the paper they cite (the 1963 "Catégories structurées") does not contain a definition of 2-category, nor the terminology. (It does introduce the notion of double category, but Ehresmann does not consider 2-categories as a special case.) It seems likely that Eilenberg and Kelly intended to cite Ehresmann's similarly named 1965 book "Catégories et Structure", in which 2-categories are mentioned. However, when I wrote to Andrée Ehresmann regarding the origin of the notion of 2-category, she wrote that Charles Ehresmann did not introduce 2-categories himself, but that they were introduced by his student Bénabou as a special kind of double category, and by different authors as categories enriched in Cat. As far as I am aware, 2-categories appeared in print for the first time in 1965, as instances of enriched categories in two different papers introducing the notion of enriched category: Bénabou's "Catégories relatives" (C. R. Acad. Sci. Paris 260), and Maranda's "Formal categories" (Canadian Journal of Mathematics 17). Consequently, my understanding is that it is appropriate to attribute the notion of 2-category jointly to Bénabou and Maranda. Best, Nathanael ________________________________ From: Ross Street <ross.street@mq.edu.au> Sent: 13 November 2023 1:58 PM To: j.adamek <j.adamek@tu-bs.de>; urs.schreiber <urs.schreiber@googlemail.com> Cc: JS Lemay <js.lemay@mq.edu.au>; Categories mailing list <categories@mq.edu.au> Subject: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions Dear Urs, Jiri, and All It is not always clear from the publication record who thought of things first. Bénabou's Intro to bicategories was published by SLNM in 1967. This is before any Oberwolfach category conference. However, it is after the La Jolla 1965 conference where Eilenberg-Kelly presented closed categories with the type-set published paper appearing in 1966. By that time we had Mac Lane's coherence theorem for monoidal categories and Kelly's reduction of the axioms to two. Bénabou's multiplicative categories tried to incorporate coherence in the definition but there is a problem with his definition. In my opinion, Bénabou's contribution was to define bicategory (using the two axiom approach) as the several object version of monoidal category, thereby initiating weak higher category theory. This took courage. John Gray was already working with 2-categories, a concept of Charles Ehresmann. John told me in 1968-9 that he was convinced of the importance of bicategories because of the example of spans in a category with pullbacks. I was already convinced by Bénabou but felt the world might not be ready for papers using them. Also in the Eilenberg-Kelly paper, there were closed and monoidal functors. They agree in the closed monoidal case. These were of the lax rather than strong kind. I understand that Eilenberg pushed the presentation of their paper into emphasising closed over monoidal categories, but both are there. In Chapter IV Section 3 on examples, they point out that a closed functor from 1 to sets is a monoid and from I into abelian groups is a ring, etc.; they recognized that monoidal functors from 1 were monoids. Bénabou's morphisms of bicategories (lax functors) became the several object version. Jack Duskin told me in 1968 that Bénabou had the construction of a 2-category C' for each category C such that lax functors out of C amounted to 2-functors out of C'. I don't know how that fits historically with the result of Lawvere for C = 1, that André Joyal mentioned, which appeared in the ``Zurich triples book'' SLNM 80 (1969). Ross

On 12 Nov 2023, at 11:13 pm, Jirí Adámek <j.adamek@tu-bs.de> wrote:

Dear All,

Bill Lawvere once told me that 'monad' had been the idea of Eilenberg. Later I asked him by email about the details and he answered the following:

I do not remember in which year it was. (Maybe 1968, judging from vague allusions in Springer Lecture Notes 80.) In any case it was in the common room of the old castle at Oberwolfach when Sammy came out from behind the piano and announced the change. His informal speech emphasized that the word would inflect well: 'monadic' etc. He also explicitly said that nobody would ever confuse it with Leibnitzian monads.

Jiri