The game of the name: Standard constructions, triples, monads, fundamental constructions

The game of the name: Standard constructions, triples, monads, fundamental constructions

[Michael Barr, Prof.]

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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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Urs Schreiber]

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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<https://protect-au.mimecast.com/s/9YEfCmO5wZsXgLMZuGXuAg?domain=ncatlab.org>

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<https://protect-au.mimecast.com/s/LtDICnx1Z5UwPpWzTJTJcd?domain=ncatlab.org>



On Thu, Nov 9, 2023 at 1:22 AM Michael Barr, Prof. <barr.michael@mcgill.ca<mailto: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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[JS Lemay]

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[[The following message is sent on behalf of posinavrayudu@gmail.com -- for whatever reason it did not seem to have been sent/approved properly to the mailing list, apologies if you receive multiple copies]]

Dear Professor Barr,

Thank you very much for sharing the history of naming and renaming
Godement's construction: la construction standarde.

If I may, having read about the influence of Godement's Théorie des
faisceaux on the work of Grothendieck
(https://www.math.mcgill.ca/barr/papers/gk.pdf<https://www.math.mcgill.ca/barr/papers/gk.pdf>) and on that of
Professor F. William Lawvere
(https://www.mat.uc.pt/~picado/lawvere/interview.pdf<https://www.mat.uc.pt/~picado/lawvere/interview.pdf>), I can't help
but wonder if there's an English translation of Godement's sheaf
theory book.

Speaking of names, unless I'm all confused about composition of
adjoint functors, another name is doctrine
(https://www.math.union.edu/~niefiels/13conference/Web/Slides/Fifty_Years_of_Functorial_Semantics.pdf<https://www.math.union.edu/~niefiels/13conference/Web/Slides/Fifty_Years_of_Functorial_Semantics.pdf>).
Please correct me if I'm mistaken.

Thanking you,
Yours truly,
posina

________________________________
From: Urs Schreiber <urs.schreiber@googlemail.com>
Sent: Thursday, November 9, 2023 3:26 PM
To: Michael Barr, Prof. <barr.michael@mcgill.ca>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

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



On Thu, Nov 9, 2023 at 1:22 AM Michael Barr, Prof. <barr.michael@mcgill.ca<mailto: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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Nathanael Arkor]

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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<https://protect-au.mimecast.com/s/oxEhC3QNl1S3yBg3fgfGcu?domain=mta.ca>), 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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Patrik Eklund]

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In our Notes for the Foundations part in our book "Semigroups in Complete Lattices" we looked into references e.g. as related to the composition of monads, which makes use of the star composition of natural transformations. That 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 règles".

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 Bénabou 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 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 generally 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 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<https://protect-au.mimecast.com/s/jDLsCNLJxkiLOz31fms6Ky?domain=mta.ca>), 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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Johannes Huebschmann]

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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<https://protect-au.mimecast.com/s/nao2Cq71jxfBEyZLSZs5CY?domain=mta.ca>), 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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Posina Venkata Rayudu]

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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<https://protect-au.mimecast.com/s/DaDVC5QP8yS8XEY8SwmXgN?domain=cahierstgdc.com>

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<https://protect-au.mimecast.com/s/sL19C6XQ68fkpzlkiNuMLK?domain=tac.mta.ca>

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<https://protect-au.mimecast.com/s/n390C71R63CM8ljMCQJ14j?domain=tac.mta.ca>) & Professor
Johnstone's Punctual Connectedness
(http://www.tac.mta.ca/tac/volumes/25/3/25-03.pdf<https://protect-au.mimecast.com/s/lZfuC81Vq2C32183H6S4n7?domain=tac.mta.ca>)

Grothendieck: Descent
(http://www.numdam.org/item/?id=SB_1958-1960__5__369_0<https://protect-au.mimecast.com/s/R03HC91W8rCnlQVntpq7co?domain=numdam.org>) & Bastiani and
Ehresmann: Sketches
(http://www.numdam.org/item/CTGDC_1972__13_2_104_0.pdf<https://protect-au.mimecast.com/s/u5Y8C0YKgRsv7V0viVH53a?domain=numdam.org>) & F. William
Lawvere: Functorial Semantics
(http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf<https://protect-au.mimecast.com/s/FVLrCgZ05Jf6o956HBLOv_?domain=tac.mta.ca>)

From my home turf of neuroscience, the Hebbian learning law: neurons
that fire together wire together, which is credited to Donald Hebb
(1949; https://drive.google.com/file/d/1_TCefN8KL36RXUA-12S3EZT57SKSoACq/view?usp=sharing<https://protect-au.mimecast.com/s/PYryCjZ12RfEB81EiBD-uk?domain=drive.google.com>,
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-diagrams/#comment-17)<https://protect-au.mimecast.com/s/qqmPCk815RClK93lIniYdz?domain=conceptualmathematics.wordpress.com>,
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-shehes/<https://protect-au.mimecast.com/s/UP_wClx1OYUGVxqGfWBPde?domain=conceptualmathematics.wordpress.com>).
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)<https://protect-au.mimecast.com/s/qA2oCmO5wZsXgzLXHZ6OiN?domain=cgasa.sbu.ac.ir>.

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<https://protect-au.mimecast.com/s/0GiGCnx1Z5UwPvpwCG3kP2?domain=ncatlab.org>
>
> 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<https://protect-au.mimecast.com/s/bw1pCoV1Y2S9gEp9fg1uTu?domain=ncatlab.org>
>
>
>
> 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
>>
>>
>>
>> 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 | Leave group | Learn more about Microsoft 365 Groups
>>
>
>
>
> 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 | Leave group | Learn more about Microsoft 365 Groups
>


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=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Learn more about Microsoft 365 Groups<https://aka.ms/o365g>


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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[JS Lemay]

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[[The following message is sent on behalf of joyal.andre@uqam.ca -- There seems to sometimes be issues with emails getting approved and sent out properly, I will try to fix this. Apologies if you receive multiple copies]]

Dear Michael,

Thank you for starting this discussion.

We all know the importance of the notion of adjunction in category theory.
The notion was introduced by Kan (1958) and  I find it surprising that it took
so long after the creation of category theory (1943).
The description of an adjunction F--| G in terms of
the adjunction identities took even longer: P.J. Huber  (1961).
I also find it surprising that there is no adjunction in Grothendieck's
Tohoku paper's (1957).
And no adjunction in Godement's "Théorie des Faiceaux"
although he introduced the notion of comonad (=construction fondamentale).

Pierre Cartier told me once (around 2015) that he and Eilenberg
had almost discovered the notion of adjoint functor before Kan.
They even published a compte-rendu (but I have not seen it).
They proved the fact the composite of two universal constructions
is universal: it amounted to showing that the composite of two left adjoint is a
left adjoint, without having defined the left adjoint from the universal constructions! I guess that they were generalising the fact
that the enveloping algebra of a free Lie algebra is a free associative algebra.
Eilenberg once told me that he had informally supervised Kan for his Phd.

The simplex category Delta was introduced by Eilenberg and Zilber,
but the notion of simplical object was then defined in terms of face
and degeneracy operators and simplicial identities, not
as a contravariant functor from Delta.
In chapter 3 of his book "Théorie des Faisceaux" Godement writes (in 3.1)
that he will not regard the sets [n]={0,....,n} as objects of a category,
because that would be too pedantic.

Best,
André
________________________________
From: Posina Venkata Rayudu <posinavrayudu@gmail.com>
Sent: Thursday, November 9, 2023 10:47 PM
To: urs.schreiber <urs.schreiber@googlemail.com>
Cc: 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

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<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<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<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<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<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<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<http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf>)

From my home turf of neuroscience, the Hebbian learning law: neurons
that fire together wire together, which is credited to Donald Hebb
(1949; https://drive.google.com/file/d/1_TCefN8KL36RXUA-12S3EZT57SKSoACq/view?usp=sharing<https://drive.google.com/file/d/1_TCefN8KL36RXUA-12S3EZT57SKSoACq/view?usp=sharing>,
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-diagrams/#comment-17)<https://conceptualmathematics.wordpress.com/2012/06/08/structure-of-internal-diagrams/#comment-17)>,
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-shehes/<https://conceptualmathematics.wordpress.com/2012/09/23/comfortable-with-shehes>).
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)<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<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<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
>>
>>
>>
>> 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 | Leave group | Learn more about Microsoft 365 Groups
>>
>
>
>
> 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 | Leave group | Learn more about Microsoft 365 Groups
>

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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Robert Pare]

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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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[JS Lemay]

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[[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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Joyal, André]

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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.

Michael



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Johannes Huebschmann]

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Dear All
Ettore Aldrovandi sent me a copy of Mac Lane's 1956 paper.
Many thanks Ettore!

In Mac Lane's [Homologie des anneaux et des modules, Louvain 1956],
the comonad and adjunction are clearly there, albeit in different
language and terminology:

In Section 3, entitled
"La méthode universelle pour les constructions standard":

- L a "subcategory" of K

- a "universal" functor T:K -> L

Being a universal functor means, in language and terminology
developed later, that the forgetful functor
F:L->K is  right adjoint to T, that is
hom(TX,Y)_L ->hom(X,FY)_K is a bijection,
for X an object of K and Y an object of L.

Now Mac Lane takes, for a discrete group Pi,

- K the category of abelian groups
- L that of Pi-modules

- T(X)= Z[Pi]otimes X

and shows how his formal construction
leads to  the standard resolution (bar resolution)
of the integers in the category L.

This is precisely the standard construction applied to
the comonad resulting from the composite functor
TF:L ->L.

Thereafter Mac Lane remarks that this construction
works for any abelian categories K and L.

The construction appears again in:
- IX.6 p. 265 ff of [Homology] (first edition 1963), with the
terminology "resolvent pair of
categories" rather than "comonad",
with a hint at the adjointness between the universal functor and the
forgetful functor,

- VII.6 (p. 181) of
[Categories for the working mathematician].
At the end of Section VI (p. 159), Mac Lane notes:
"Mac Lane [1956] mentioned in passing (his §3) that all the standard
resolutions could be obtained from universal arrows (i.e., from
adjunctions). Then Godement [1958] systematized these resolutions
by using standard constructions (comonads)."
I guess Mac Lane here actually means
"using dual standard constructions (monads)".

Godement [1958] does not contain a bibliography
and cites only a few papers within the text.
Eilenberg-Mac Lane show up but, as far as I can see,
without explicit mention of any of their papers.
Also, Godement does not claim any originality.
The first phrase of the introduction reads:
"De toutes les idées qui circulent dans les milieux mathématiques
actuels, celle de publier un ouvrage de référence consacré
à le théorie des faisceaux et assûrément l'une des moins originales."

According to MR, Godement did not publish a paper in
[Colloque de topologie algébrique, Louvain, 1956],
and I do not know whether he attended the meeting.

BTW from what Ettore Aldrovandi
sent me, I also learnt that F. Adams attributes the terminology
"cobar construction" to H. Cartan.

Best

Johannes


________________________________
De: "Joyal, André" <joyal.andre@uqam.ca>
À: "Johannes Huebschmann" <johannes.huebschmann@univ-lille.fr>, "Nathanael Arkor" <nathanael.arkor@gmail.com>
Cc: "categories" <categories@mq.edu.au>
Envoyé: Jeudi 9 Novembre 2023 19:07:58
Objet: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

Dear Michael,

Thank you for starting this discussion.

We all know the importance of the notion of adjunction in category theory.
The notion was introduced by Kan (1958) and  I find it surprising that it took
so long after the creation of category theory (1943).
The description of an adjunction F--| G in terms of
the adjunction identities took even longer: P.J. Huber  (1961).
I also find it surprising that there is no adjunction in Grothendieck's
Tohoku paper's (1957).
And no adjunction in Godement's "Théorie des Faiceaux"
although he introduced the notion of comonad (=construction fondamentale).

Pierre Cartier told me once (around 2015) that he and Eilenberg
had almost discovered the notion of adjoint functor before Kan.
They even published a compte-rendu (but I have not seen it).
They proved the fact the composite of two universal constructions
is universal: it amounted to showing that the composite of two left adjoint is a
left adjoint, without having defined the left adjoint from the universal constructions! I guess that they were generalising the fact
that the enveloping algebra of a free Lie algebra is a free associative algebra.
Eilenberg once told me that he had informally supervised Kan for his Phd.

The simplex category Delta was introduced by Eilenberg and Zilber,
but the notion of simplical object was then defined in terms of face
and degeneracy operators and simplicial identities, not
as a contravariant functor from Delta.
In chapter 3 of his book "Théorie des Faisceaux" Godement writes (in 3.1)
that he will not regard the sets [n]={0,....,n} as objects of a category,
because that would be too pedantic.

Best,
André







________________________________
De : Johannes Huebschmann <johannes.huebschmann@univ-lille.fr>
Envoyé : 9 novembre 2023 04:29
À : Nathanael Arkor <nathanael.arkor@gmail.com>
Cc : categories@mq.edu.au <categories@mq.edu.au>; Johannes Huebschmann <johannes.huebschmann@univ-lille.fr>
Objet : Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

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<https://protect-au.mimecast.com/s/twAlCQnM1WfVwVErhxwFQX?domain=mta.ca>), 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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Urs Schreiber]

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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



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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Jirí Adámek]

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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Ross Street]

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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Michael Barr, Prof.]

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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>
Sent: Sunday, November 12, 2023 9:58 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
>



----------

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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Ross Street]

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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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[JS Lemay]

[-- Attachment #1: Type: text/plain, Size: 4835 bytes --]

[[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
>


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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[John Baez]

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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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Ross Street]

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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Richard Garner]

Earlier on Mac Lane did also use "bicategory" to mean more or less what
we would nowadays call "category with a proper factorisation system".
(Starting already in 1950 in "Duality for groups".) Isbell used this
terminology a lot and as late as 1964.

Richard

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

> 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
>> 
>> 

Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Dusko Pavlovic]

>> 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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Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[dawson]

On 2023-11-14 17:11, Dusko Pavlovic wrote:

> so it is a little ironic to find on the other end the history of
> mathematics being the tricky part.

Oh, I don't know! This interesting and important question seems to have
taken a few thousand words and a few person-hours to resolve, apparently
to everybody's satisfaction. Many a question in (for instance) category
theory requires (at least) twice as many words, ten times the time, and
a bundle of commutative diagrams.

But it's not a contest. We can (and should) do both.

Best wishes to all,
Robert







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