[[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

which made me think it would be nice to reprint seminal unpublished
works of Professor Benabou (such as those discussed in the above) as:

Reprints in Theory and Applications of Categories
http://www.tac.mta.ca/tac/reprints/index.html

simply because Professor Benabou's orginal conceptualization of many
category theoretic constructs, beginning with closed to fibered
categories, along with, of course, monads/triples (ibid., ref. 12),
are worth studying in and of themselves and/or in the context of thier
conceptual cousins, so to speak. Here are a couple of illustrations
of conceptual kinship that is quite commonplace in science:

Professor F. William Lawvere's Axiomatic Cohesion
(http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf) & Professor
Johnstone's Punctual Connectedness
(http://www.tac.mta.ca/tac/volumes/25/3/25-03.pdf)

Grothendieck: Descent
(http://www.numdam.org/item/?id=SB_1958-1960__5__369_0) & Bastiani and
Ehresmann: Sketches
(http://www.numdam.org/item/CTGDC_1972__13_2_104_0.pdf) & F. William
Lawvere: Functorial Semantics
(http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf)

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,
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),
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/).
Here's one direction to move on i.e., build on Leibniz monad to get to
intensive quality (e.g., idempotent;
https://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.pdf).

Thanking you,
Yours truly,
posina


On Thu, Nov 9, 2023 at 12:44 PM Urs Schreiber
<urs.schreiber@googlemail.com> wrote:
>
> Just to note that in 2009 on this same list, the lunch-genesis of "monad" was attributed to Jean Benabou, see
>
> Barr 2009 https://ncatlab.org/nlab/show/monad#Barr09
>
> Indeed, in print the term was introduced by
>
> Benabou 1967: "Introduction to Bicategories" (section 5.4)
>
> together with the astute observation that monads are the lax images of 1
> and thus quite the 2-categorical version of the units=monads of Euclid.
>
> https://ncatlab.org/nlab/show/monad+terminology
>
>
>
> On Thu, Nov 9, 2023 at 1:22 AM Michael Barr, Prof. <barr.michael@mcgill.ca> wrote:
>>
>> People seemed to enjoy my history of the founding of TAC, so I thought you might enjoy my sharing of other historical notes.
>>
>> This construction was introduced in Godement's book Théorie des faisceaux in connection with his resolution of sheaves by "faisceaux mous" (soft sheaves) which are an injective class. He called this "la construction standarde". It is not clear whether this was intended to name them or merely describe them. At any rate, around 1960,
>>
>> Benno Eckmann and his students took as a name and called them standard constructions. One of the students, Peter Huber, told me that they were having trouble, in particular cases, verifying the associative law. And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U. He wondered if every adjoint pair gave rise to a standard construction and proved that it did. Then another student, Heinrich Kleisli, showed that the converse was also true. That gave us the well-known Kleisli construction.
>>
>> In 1963 Samuel Eilenberg and John Moore published a monograph called Foundations of Relative Homological Algebra in which they used this construction as basic. Only they didn't call them standard constructions; they called them triples. I once asked Sammy why and he replied that it didn't seem like an important concept and it didn't seem worth it to spend a lot of time worrying about the name. This is in stark contrast with the time he and Henri Cartan spent thinking about the name for their basic sequences. There is a story, perhaps apocryphal, that their book was in proof stage before they settled on the exact name.
>>
>> So triple was name Jon Beck and I were using in our joint work on homological algebra. Then in 1966 there was a category meeting in Oberwohlfach and there was a lot of discussion of a better name. The next bit of the story comes out of my extremely fallible memory and could well be mistaken. One day at lunch I was sitting next to Anders Koch and he asked what I thought about the name monad. I thought (and still think!) it was a pretty good name and so he proposed it and the assembled crowd agreed and adopted it. I would have too, but Jon rejected it. Why, I asked him. He did not think it a good name and refused to use it. He said there was no point in replacing one bad name by another. Since we were collaborating and since he was even more stubborn than me, that's they way it was. In our papers, Jon insisted on putting functions to the right of their arguments, just like reverse Polish.
>>
>> Then we stopped collaborating and, by 1980, I think I was about ready to start using monad. But then TTT came along and the alliteration was just too good to pass up. Charles Wells agreed on those grounds.
>>
>> And what about fundamental construction? I spent six and a half months at the ETH in Zurich. A few days after I arrived, I got a phone call from Peter Huber, the aforementioned former student of Eckmann's. He had just received from Math Reviews a paper written by Jean-Marie Maranda that used that term for the concept and Huber asked me if there was any way to stop that proliferation of names. As far as I know, that was the only place that term was ever used.
>>
>> Michael
>>
>>
>>
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