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charset="utf-8" Content-Transfer-Encoding: quoted-printable 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 m=C3=A9thode 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)=3D 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 =C2=A73) 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 id=C3=A9es qui circulent dans les milieux math=C3=A9matiques actuels, celle de publier un ouvrage de r=C3=A9f=C3=A9rence consacr=C3=A9 =C3=A0 le th=C3=A9orie des faisceaux et ass=C3=BBr=C3=A9ment l'une des moin= s originales." According to MR, Godement did not publish a paper in [Colloque de topologie alg=C3=A9brique, 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, Andr=C3=A9" =C3=80: "Johannes Huebschmann" , "Natha= nael Arkor" Cc: "categories" Envoy=C3=A9: Jeudi 9 Novembre 2023 19:07:58 Objet: Re: The game of the name: Standard constructions, triples, monads, f= undamental 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 t= ook 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 "Th=C3=A9orie des Faiceaux" although he introduced the notion of comonad (=3Dconstruction 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 co= nstructions! I guess that they were generalising the fact that the enveloping algebra of a free Lie algebra is a free associative alg= ebra. 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 "Th=C3=A9orie des Faisceaux" Godement writes (in 3= .1) that he will not regard the sets [n]=3D{0,....,n} as objects of a category, because that would be too pedantic. Best, Andr=C3=A9 ________________________________ De : Johannes Huebschmann Envoy=C3=A9 : 9 novembre 2023 04:29 =C3=80 : Nathanael Arkor Cc : categories@mq.edu.au ; Johannes Huebschmann 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" - r=C3=A9solution 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 constructio= n. 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 mona= d), 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 "r=C3=A9solution simpliciale = standard". Best Johannes ________________________________ De: "Nathanael Arkor" =C3=80: categories@mq.edu.au Envoy=C3=A9: Jeudi 9 Novembre 2023 07:03:42 Objet: Re: The game of the name: Standard constructions, triples, monads, f= undamental constructions Dear Michael, Thank you very much for sharing this piece of history. I, personally, am al= ways deeply interested to learn of the stories behind the mathematics, and = the people responsible for it, that have been fundamental in shaping what w= e understand to be category theory today. In the interest of posterity, I have a slight amendment to the etymology yo= u described. I should preface what follows by making it clear that my under= standing is based only on the literature, rather than personal experience, = and so may not accurately reflect the actual history (in which case I would= be glad to be corrected). (On fundamental/standard constructions) The terminology employed by Godement (on p. 271 of the 1958 "Topologie alg= =C3=A9brique et th=C3=A9orie des faisceaux") was "la construction fondament= ale", i.e. "the fundamental construction". It was thus Godement rather than= Maranda who introduced this terminology; Maranda appears to be the only on= e who continued to use that terminology in later work. Huber then employed = the terminology "standard construction" for the notion of comonad (in =C2= =A72 of the 1961 "Homotopy Theory in General Categories") =E2=80=93 monads = are instead called "dual standard constructions". It does not appear to be = until around 1968 that the terminology "standard construction" appears in r= eference to the concept of monad rather than comonad. (The seemingly misinf= ormed assertion that Godement introduced the terminology "standard construc= tion" appears, perhaps for the first time, in the 1969 Proceedings of "Semi= nar on Triples and Categorical Homology Theory", which may be where confusi= on has arisen.) (On triples) I believe the paper of Eilenberg and Moore in which the terminology "triple= " first appears is the 1965 "Adjoint functors and triples" (I did not find = the terminology "triple" in "Foundations of Relative Homological Algebra").= (Incidentally, in Dubuc's 1968 paper "Adjoint triangles", he refers to the= terminology "triplex", but this appears to be a typo, as I cannot find tha= t terminology elsewhere.) (On monads) In a 2009 email to the categories mailing list (https://www.mta.ca/~cat-dis= t/archive/2009/09-4), you recounted the same story about the origin of the t= erminology "monad", except that you recalled the one who proposed the termi= nology was Jean B=C3=A9nabou. This seems likely, since, as far as I can tel= l, the term first appears in B=C3=A9nabou's 1967 "Introduction to bicategor= ies" (p. 39), where the terminology is justified in a footnote on p. 40. Best, Nathanael On 8 Nov 2023, 22:22 +0100, Michael Barr, Prof. , w= rote: People seemed to enjoy my history of the founding of TAC, so I thought you = might enjoy my sharing of other historical notes. This construction was introduced in Godement's book Th=C3=A9orie des faisce= aux in connection with his resolution of sheaves by "faisceaux mous" (soft = sheaves) which are an injective class. He called this "la construction sta= ndarde". It is not clear whether this was intended to name them or merely = describe them. At any rate, around 1960, Benno Eckmann and his students took as a name and called them standard con= structions. One of the students, Peter Huber, told me that they were havin= g trouble, in particular cases, verifying the associative law. And then he= noticed that in all the cases he knew, the functor T had the form UF, wher= e F --| U. He wondered if every adjoint pair gave rise to a standard const= ruction and proved that it did. Then another student, Heinrich Kleisli, sh= owed that the converse was also true. That gave us the well-known Kleisli = construction. In 1963 Samuel Eilenberg and John Moore published a monograph called Founda= tions of Relative Homological Algebra in which they used this construction = as basic. Only they didn't call them standard constructions; they called t= hem triples. I once asked Sammy why and he replied that it didn't seem lik= e an important concept and it didn't seem worth it to spend a lot of time w= orrying about the name. This is in stark contrast with the time he and Hen= ri Cartan spent thinking about the name for their basic sequences. There i= s a story, perhaps apocryphal, that their book was in proof stage before th= ey settled on the exact name. So triple was name Jon Beck and I were using in our joint work on homologic= al algebra. Then in 1966 there was a category meeting in Oberwohlfach and = there was a lot of discussion of a better name. The next bit of the story = comes out of my extremely fallible memory and could well be mistaken. One = day at lunch I was sitting next to Anders Koch and he asked what I thought = about the name monad. I thought (and still think!) it was a pretty good na= me and so he proposed it and the assembled crowd agreed and adopted it. I = would have too, but Jon rejected it. Why, I asked him. He did not think i= t a good name and refused to use it. He said there was no point in replaci= ng one bad name by another. Since we were collaborating and since he was e= ven more stubborn than me, that's they way it was. In our papers, Jon insi= sted on putting functions to the right of their arguments, just like revers= e Polish. Then we stopped collaborating and, by 1980, I think I was about ready to st= art using monad. But then TTT came along and the alliteration was just too= good to pass up. Charles Wells agreed on those grounds. And what about fundamental construction? I spent six and a half months at = the ETH in Zurich. A few days after I arrived, I got a phone call from Pet= er Huber, the aforementioned former student of Eckmann's. He had just rece= ived from Math Reviews a paper written by Jean-Marie Maranda that used that= term for the concept and Huber asked me if there was any way to stop that = proliferation of names. As far as I know, that was the only place that ter= m was ever used. Michael You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --=_deeb1872-4db0-41fb-a9eb-c8be6a4df18c Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
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 m=C3=A9thode 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)=3D 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 =C2=A73) that all the stand= ard
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 id=C3=A9es qui circulent dans les milieux math=C3=A9mat= iques
actuels, celle de publier un ouvrage de r=C3=A9f=C3=A9rence consacr=C3=A9 <= br> =C3=A0 le th=C3=A9orie des faisceaux et ass=C3=BBr=C3=A9ment l'une des moin= s originales."

According to MR, Godement did not publish a paper in
[Colloque de topologie alg=C3=A9brique, 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, Andr=C3=A9" &= lt;joyal.andre@uqam.ca>
=C3=80: "Johannes Huebschmann" <johannes.huebschmann@un= iv-lille.fr>, "Nathanael Arkor" <nathanael.arkor@gmail.com&= gt;
Cc: "categories" <categories@mq.edu.au>
Envoy=C3=A9: Jeudi 9 Novembre 2023 19:07:58
Objet: Re: The game of the name: Standard constructions, triples, mo= nads, fundamental constructions

<= /div>
Dear Michael, 

Thank you for starting this discussion.

We all know the importance of the notion of adju= nction 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 adjun= ction in Grothendieck's
Tohoku paper's (1957).
And no adjunction in Godement's "Th=C3=A9or= ie des Faiceaux"
although he introduced the notion of comonad (= =3Dconstruction fondamentale).

Pierre Cartier told me once (around 2015) that h= e and Eilenberg
had almost discovered the notion of adjoint func= tor before Kan.
They even published a compte-rendu (but I have n= ot seen it). 
They proved the fact the composite of two univer= sal constructions
is universal: it amounted to showing that the co= mposite of two left adjoint is a
left adjoint, without having defined the left ad= joint from the universal constructions! I guess that they were generalising the fact
that the enveloping algebra of a free Lie algebr= a is a free associative algebra.
Eilenberg once told me that he had informally su= pervised Kan for his Phd.

The simplex category Delta was introduced by Eil= enberg and Zilber,
but the notion of simplical object was then defi= ned in terms of face
and degeneracy operators and simplicial identiti= es, not
as a contravariant functor from Delta.
In chapter 3 of his book "Th=C3=A9orie des = Faisceaux" Godement writes (in 3.1)
that he will not regard the sets [n]=3D{0,....,n= } as objects of a category,
because that would be too pedantic.

Best,
Andr=C3=A9







De : Johannes Huebschmann &= lt;johannes.huebschmann@univ-lille.fr>
Envoy=C3=A9 : 9 novembre 2023 04:29
=C3=80 : Nathanael Arkor <nathanael.arkor@gmail.com>
Cc : categories@mq.edu.au <categories@mq.edu.au>; Johanne= s Huebschmann <johannes.huebschmann@univ-lille.fr>
Objet : Re: The game of the name: Standard constructions, triples, m= onads, 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 cate= gories,
thereby giving the standard constructions of homological algebra"= .



Godement's terminology [1958] p. 270/71:

- "ss object" ("objet semi-simplicial") for nowaday= s "cosimplicial object"

- r=C3=A9solution simpliciale standard: a cosimplicial object which yi= elds 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 termino= logy "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 "m= onade"
in his book "La monadologie".

In  French, the correct wording is
"construction standard", cf. above Godement's "r=C3=A9s= olution simpliciale standard".


Best

Johannes




De: "Nathanael Arkor" <nathanael.arkor@gmail.com&g= t;
=C3=80: categories@mq.edu.au
Envoy=C3=A9: Jeudi 9 Novembre 2023 07:03:42
Objet: Re: The game of the name: Standard constructions, triples, mo= nads, fundamental constructions

Dear Michael,

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

In the interest of posterity, I have a slight amendment to the etymology yo= u described. I should preface what follows by making it clear that my under= standing is based only on the literature, rather than personal experience, = and so may not accurately reflect the actual history (in which case I would be glad to be corrected).

(On fundamental/standard constructions)
The terminology employed by Godement (on p. 271 of the 1958 "Topologie= alg=C3=A9brique et th=C3=A9orie des faisceaux") was "la construc= tion fondamentale", i.e. "the fundamental construction". It = was thus Godement rather than Maranda who introduced this terminology; Mara= nda appears to be the only one who continued to use that terminology in later = work. Huber then employed the terminology "standard construction"= for the notion of comonad (in =C2=A72 of the 1961 "Homotopy Theory in= General Categories") =E2=80=93 monads are instead called "dual standard constructions". It does not appear to be until around 1968 t= hat the terminology "standard construction" appears in reference = to the concept of monad rather than comonad. (The seemingly misinformed ass= ertion that Godement introduced the terminology "standard construction" appears, perhaps for the first time, in = the 1969 Proceedings of "Seminar on Triples and Categorical Homology T= heory", which may be where confusion has arisen.)

(On triples)
I believe the paper of Eilenberg and Moore in which the terminology "t= riple" first appears is the 1965 "Adjoint functors and triples&qu= ot; (I did not find the terminology "triple" in "Foundations= of Relative Homological Algebra"). (Incidentally, in Dubuc's 1968 paper "Adjoint triangles", he refers to the terminology "tr= iplex", but this appears to be a typo, as I cannot find that terminolo= gy elsewhere.)

(On monads)
In a 2009 email to the categories mailing list (https://www.mta.ca/~cat-dist/archive/2009/09-4), you recounted the sam= e story about the origin of the terminology "monad", except that you recalled the one who proposed the termi= nology was Jean B=C3=A9nabou. This seems likely, since, as far as I can tel= l, the term first appears in B=C3=A9nabou's 1967 "Introduction to bica= tegories" (p. 39), where the terminology is justified in a footnote on p. 40.

Best,
Nathanael
On 8 Nov 2023, 22:22 +0100, Michael Barr, Prof. <barr.michael@mcgil= l.ca>, wrote:
People seemed to enjoy my history of the founding of TAC, so I thought you = might enjoy my sharing of other historical notes.

This construction was introduced in Godement's book Th=C3=A9orie des faisce= aux in connection with his resolution of sheaves by "faisceaux mous&qu= ot; (soft sheaves) which are an injective class.  He called this "= ;la construction standarde".  It is not clear whether this was intended to name them or merely describe them.  At any rate, arou= nd 1960,

 Benno Eckmann and his students took as a name and called them standar= d constructions.  One of the students, Peter Huber, told me that they = were having trouble, in particular cases, verifying the associative law.&nb= sp; And then he noticed that in all the cases he knew, the functor T had the form UF, where F --| U.  He wondered if e= very adjoint pair gave rise to a standard construction and proved that it d= id.  Then another student, Heinrich Kleisli, showed that the converse = was also true.  That gave us the well-known Kleisli construction. 

In 1963 Samuel Eilenberg and John Moore published a monograph called Founda= tions of Relative Homological Algebra in which they used this construction = as basic.  Only they didn't call them standard constructions; they cal= led them triples.  I once asked Sammy why and he replied that it didn't seem like an important concept and it di= dn't seem worth it to spend a lot of time worrying about the name.  Th= is is in stark contrast with the time he and Henri Cartan spent thinking ab= out the name for their basic sequences.  There is a story, perhaps apocryphal, that their book was in proof stage b= efore they settled on the exact name.

So triple was name Jon Beck and I were using in our joint work on homologic= al algebra.  Then in 1966 there was a category meeting in Oberwohlfach= and there was a lot of discussion of a better name.  The next bit of = the story comes out of my extremely fallible memory and could well be mistaken.  One day at lunch I was sitting ne= xt to Anders Koch and he asked what I thought about the name monad.  I= thought (and still think!) it was a pretty good name and so he proposed it= and the assembled crowd agreed and adopted it.  I would have too, but Jon rejected it.  Why, I asked him.&n= bsp; He did not think it a good name and refused to use it.  He said t= here was no point in replacing one bad name by another.  Since we were= collaborating and since he was even more stubborn than me, that's they way it was.  In our papers, Jon insisted on putting funct= ions to the right of their arguments, just like reverse Polish. 

Then we stopped collaborating and, by 1980, I think I was about ready to st= art using monad.  But then TTT came along and the alliteration was jus= t too good to pass up.  Charles Wells agreed on those grounds.

And what about fundamental construction?  I spent six and a half month= s at the ETH in Zurich.  A few days after I arrived, I got a phone cal= l from Peter Huber, the aforementioned former student of Eckmann's.  H= e had just received from Math Reviews a paper written by Jean-Marie Maranda that used that term for the concept and Huber asked = me if there was any way to stop that proliferation of names.  As far a= s I know, that was the only place that term was ever used.

Michael

 
 
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