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charset="utf-8"; format=flowed Content-Transfer-Encoding: quoted-printable In our Notes for the Foundations part in our book "Semigroups in Complete L= attices" we looked into references e.g. as related to the composition of mo= nads, which makes use of the star composition of natural transformations. T= hat construction, as we write, seems to go back to Ehresmann 1960, and the = star composition can also be seen already in the appendix of Godement 1958 = in his "Cinq r=C3=A8gles". Huber 1961 indeed investigated the relationship between adjoint situations = and monads, as Michael and Nathanael write, and some years later, Eilenberg= , Moore and Kleisli along those lines. Monoidal categories go back to 1963, with B=C3=A9nabou and Mac Lane. --- The history of the term functor/monad I also find interesting.as it relates= to the use of underlying signatures, and in particular when many-sorted si= gnatures are used. That history is not all that clear to me. We provided co= nstructions after 2010, as we needed them e.g. in applications using monad = compositions, and more generally for term monads over any monoidal closed c= ategory. Best, Patrik On 2023-11-09 08:03, Nathanael Arkor wrote: Dear Michael, Thank you very much for sharing this piece of history. I, personally, am al= ways deeply interested to learn of the stories behind the mathematics, and = the people responsible for it, that have been fundamental in shaping what w= e understand to be category theory today. In the interest of posterity, I have a slight amendment to the etymology yo= u described. I should preface what follows by making it clear that my under= standing 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 alg= =C3=A9brique et th=C3=A9orie des faisceaux") was "la construction fondament= ale", i.e. "the fundamental construction". It was thus Godement rather than= Maranda who introduced this terminology; Maranda appears to be the only on= e who continued to use that terminology in later work. Huber then employed = the terminology "standard construction" for the notion of comonad (in =C2= =A72 of the 1961 "Homotopy Theory in General Categories") =E2=80=93 monads = are instead called "dual standard constructions". It does not appear to be = until around 1968 that the terminology "standard construction" appears in r= eference to the concept of monad rather than comonad. (The seemingly misinf= ormed assertion that Godement introduced the terminology "standard construc= tion" appears, perhaps for the first time, in the 1969 Proceedings of "Semi= nar on Triples and Categorical Homology Theory", which may be where confusi= on 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 tha= t terminology elsewhere.) (On monads) In a 2009 email to the categories mailing list (https://www.mta.ca/~cat-dis= t/archive/2009/09-4), you recounted the same story about the origin of the t= erminology "monad", except that you recalled the one who proposed the termi= nology was Jean B=C3=A9nabou. This seems likely, since, as far as I can tel= l, the term first appears in B=C3=A9nabou's 1967 "Introduction to bicategor= ies" (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. , w= rote: 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 Th=C3=A9orie des faisce= aux in connection with his resolution of sheaves by "faisceaux mous" (soft = sheaves) which are an injective class. He called this "la construction sta= ndarde". 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 con= structions. One of the students, Peter Huber, told me that they were havin= g 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, wher= e F --| U. He wondered if every adjoint pair gave rise to a standard const= ruction and proved that it did. Then another student, Heinrich Kleisli, sh= owed 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 Founda= tions of Relative Homological Algebra in which they used this construction = as basic. Only they didn't call them standard constructions; they called t= hem triples. I once asked Sammy why and he replied that it didn't seem lik= e an important concept and it didn't seem worth it to spend a lot of time w= orrying about the name. This is in stark contrast with the time he and Hen= ri Cartan spent thinking about the name for their basic sequences. There i= s a story, perhaps apocryphal, that their book was in proof stage before th= ey settled on the exact name. So triple was name Jon Beck and I were using in our joint work on homologic= al 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 na= me 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 i= t a good name and refused to use it. He said there was no point in replaci= ng one bad name by another. Since we were collaborating and since he was e= ven more stubborn than me, that's they way it was. In our papers, Jon insi= sted on putting functions to the right of their arguments, just like revers= e Polish. Then we stopped collaborating and, by 1980, I think I was about ready to st= art 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 Pet= er Huber, the aforementioned former student of Eckmann's. He had just rece= ived 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 ter= m was ever used. Michael You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --=_89f6ff6be60037ee5588cef032e688d1 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8

In our Notes for the Foundations part in our book "Semigroups in Co= mplete Lattices" we looked into references e.g. as related to the comp= osition of monads, which makes use of the star composition of natural trans= formations. That construction, as we write, seems to go back to Ehresmann 1960, and the star composition can also be s= een already in the appendix of Godement 1958 in his "Cinq r=C3=A8gles&= quot;.

Huber 1961 indeed investigated the relationship between adjoint situatio= ns and monads, as Michael and Nathanael write, and some years later, Eilenb= erg, Moore and Kleisli along those lines.

Monoidal categories go back to 1963, with B=C3=A9nabou and Mac Lane.

---

The history of the term functor/monad I also find interesting.as it rela= tes to the use of underlying signatures, and in particular when many-sorted= signatures are used. That history is not all that clear to me. We provided= constructions after 2010, as we needed them e.g. in applications using monad compositions, and more genera= lly for term monads over any monoidal closed category.

Best,

Patrik


On 2023-11-09 08:03, Nathanael Arkor wrote:

Dear Michael,

Thank you very much for sharing this piece of history. I, personally, am al= ways deeply interested to learn of the stories behind the mathematics, and = the people responsible for it, that have been fundamental in shaping what w= e understand to be category theory today.

In the interest of posterity, I have a slight amendment to the etymology yo= u described. I should preface what follows by making it clear that my under= standing 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= alg=C3=A9brique et th=C3=A9orie des faisceaux") was "la construc= tion fondamentale", i.e. "the fundamental construction". It = was thus Godement rather than Maranda who introduced this terminology; Mara= nda 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 =C2=A72 of the 1961 "Homotopy Theory in= General Categories") =E2=80=93 monads are instead called "dual standard constructions". It does not appear to be until around 1968 t= hat the terminology "standard construction" appears in reference = to the concept of monad rather than comonad. (The seemingly misinformed ass= ertion that Godement introduced the terminology "standard construction" appears, perhaps for the first time, in = the 1969 Proceedings of "Seminar on Triples and Categorical Homology T= heory", which may be where confusion has arisen.)

(On triples)
I believe the paper of Eilenberg and Moore in which the terminology "t= riple" first appears is the 1965 "Adjoint functors and triples&qu= ot; (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 "tr= iplex", but this appears to be a typo, as I cannot find that terminolo= gy 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 B=C3=A9nabou. This seems like= ly, since, as far as I can tell, the term first appears in B=C3=A9nabou's 1= 967 "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@mcgil= l.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 Th=C3=A9orie des faisce= aux in connection with his resolution of sheaves by "faisceaux mous&qu= ot; (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, arou= nd 1960,
 
 Benno Eckmann and his students took as a name and called them standar= d constructions.  One of the students, Peter Huber, told me that they = were having trouble, in particular cases, verifying the associative law.&nb= sp; And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U.  He wondered if e= very adjoint pair gave rise to a standard construction and proved that it d= id.  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 Founda= tions of Relative Homological Algebra in which they used this construction = as basic.  Only they didn't call them standard constructions; they cal= led them triples.  I once asked Sammy why and he replied that it didn't seem like an important concept and it di= dn't seem worth it to spend a lot of time worrying about the name.  Th= is is in stark contrast with the time he and Henri Cartan spent thinking ab= out the name for their basic sequences.  There is a story, perhaps apocryphal, that their book was in proof stage b= efore they settled on the exact name.
 
So triple was name Jon Beck and I were using in our joint work on homologic= al 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 ne= xt 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.&n= bsp; He did not think it a good name and refused to use it.  He said t= here 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 funct= ions to the right of their arguments, just like reverse Polish. 
 
Then we stopped collaborating and, by 1980, I think I was about ready to st= art using monad.  But then TTT came along and the alliteration was jus= t too good to pass up.  Charles Wells agreed on those grounds.
 
And what about fundamental construction?  I spent six and a half month= s at the ETH in Zurich.  A few days after I arrived, I got a phone cal= l from Peter Huber, the aforementioned former student of Eckmann's.  H= e 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 a= s 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 mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
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