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

[Johannes Huebschmann <johannes.huebschmann@univ-lille.fr>]

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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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