Dear All, Bill Lawvere has an important contribution to the theory of monads. He showed that the category Delta(+) of finite ordinals and order preserving maps is monoidal and freely generated by a monoid object. It is the best possible explanation for the fact (discovered by Godement and Huber) that you can define a cosimplicial object from a monad. Special cases of that were known before, with the bar-construction of an associative algebra (the bar construction was invented by Eilenberg and MacLane). André ________________________________ De : JS Lemay <js.lemay@mq.edu.au> Envoyé : 9 novembre 2023 15:37 À : Michael Barr, Prof. <barr.michael@mcgill.ca>; Categories mailing list <categories@mq.edu.au> Objet : Re: The game of the name: Standard constructions, triples, monads, fundamental constructions [[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. 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