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Email Gateway (Nov 08 2023 at 21:22):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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Email Gateway (Nov 09 2023 at 05:09):Just to note that in 2009 on this same list, the lunch-genesis of "monad" was attributed to Jean Benabou, see
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.
Email Gateway (Nov 09 2023 at 05:41):[[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
Email Gateway (Nov 09 2023 at 06:06):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
Email Gateway (Nov 09 2023 at 08:38):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
Email Gateway (Nov 09 2023 at 09:38):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
Email Gateway (Nov 09 2023 at 19:12):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
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Email Gateway (Nov 09 2023 at 19:32):[[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
Email Gateway (Nov 09 2023 at 20:10):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
Email Gateway (Nov 09 2023 at 20:39):[[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
Email Gateway (Nov 09 2023 at 21:11):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