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charset="utf-8" Content-Transfer-Encoding: quoted-printable Isn't this idea of reconstructing G from G/N-->Aut(N)/Out(N) an early antic= ipation of Yoneda methods? Amazing. Or am I projecting? In any case, thank = you for a beautiful story beautifully told! But don't you think that Turing would disagree with the first sentence of y= our article? Chatbots will need a while before they can catch up with being= as much of a threat to the health of anything as people are. Best wishes, -- dusko On Mon, Jan 1, 2024 at 1:13=E2=80=AFPM Johannes Huebschmann > wrote: Dear All Among many items, the EML collaboration developed, in terms of a 3-cohomology class, the obstruction to solving the extension problem for non-abelian groups. This obstruction is implicit in a paper by A. Turing published nine years before EML's paper. Details are in https://www.ams.org/journals/notices/202311/rnoti-p1802.pdf Best wishes for the new year. Johannes ________________________________ De: "Joyal, Andr=C3=A9" > =C3=80: "Michael Barr, Prof." >, "categories" = > Envoy=C3=A9: Mardi 26 D=C3=A9cembre 2023 00:56:28 Objet: Re: History: The Eilenberg-Mac Lane collaboration Dear Michael, Thank you for your recollection of collaboration between Sammy and Saunders= . You wrote: They spent five years publishing their work from the war years and then t= heir collaboration ceased. Almost surely because they were no longer in th= e same location. Mail was slow and unsatisfactory and there was no interne= t. When Charles and I were trying to collaborate on TTT, mail between Mont= real and Cleveland took a minimum of two weeks. Then we both got computers= and we discovered data transfers and we were off. Of course, collaboratio= n by mail was possible, but highly unsatisfactory. I do not have a direct knowledge of this, but Myles Tierney told me another= story a few decades ago. As you know, Steenrod discovered his (mod 2) cohomology operations in 1947. Eilenberg and MacLane saw that the operations could be explained in terms o= f the mod 2 cohomology of the Eilenberg-MacLane spaces K(Z/2, n). They decided to compute the cohomology of K(A,n) for any abelian group A and any n >0. They published three notes on the cohomology theory of abelian groups in the Proceedings of the National Academy of Sciences (1950). Followed by three papers on the group H(pi,n) in 1953. Cartan was reading their work closely, and he began to work on the problem with Serre in the early 50. Eilenberg and MacLane had been trying to solve the general problem by iterating some general contruction (the famous bar resolution). In contrast, Cartan and Serre concentrated their effort on the special case of K(Z/p,n) for a prime p. Also, they exploited systematically the fact that K(A, n) is the base space= of a fibration with a contractible total space E(A,n) and fiber K(A,n-1). Eilenberg evetually understood that Cartan's approach was more powerful and after visiting him in Paris 1952 he started a collaboration. I guess that their Homological Algebra is one upshot. According to Myles, MacLane was upset and he never worked with Eilenberg ag= ain. He authored a book on homological algebra using abelian categories years la= ter. Fortunately, Eilenberg and MacLane stayed in contact: they were the founding fathers of the growing school of category theory! The integral homology of K(pi,n) was computed in general by G.J. Decker in his Phd thesis (1974) with MacLane, but his description is very complica= ted. Some finite abelian groups are only described with an infinite presentation= ! For a better description in special cases, see the paper of Larry Breen, R. Mikhailov and A. Touze (2013). > Larry was planning to describe them all explicitly. Unfortunately, he died May 8, 2023 (after having Alzheimer for many years)= . He was a very good mathematician. http://www.ams.org/news?news_id=3D7249 News from the AMS Advancing research. Creating connections. www.ams.org Best wishes, Andr=C3=A9 ________________________________ De : Michael Barr, Prof. > Envoy=C3=A9 : 17 d=C3=A9cembre 2023 21:19 =C3=80 : categories@mq.edu.au > Objet : History: The Eilenberg-Mac Lane collaboration I would like to tell some of the story of Eilenberg-Mac Lane. I will start= with a tale told me my a graduate student at Columbia when I began as an i= nstructor in 1962. He said that Sammy had never met Saunders in person until the International= Congress in Cambridge, Mass in 1950 and that after they met, they never co= llaborated again. It is hard to imagine a more wrong story, but the student appeared to belie= ve it. In fact, Sammy arrived in the US in (I could be off by a year) 1939 and got= a one year appointment at U. Michigan. At that time he was working on the= (co)homology of K(pi,1). A space with only one non-zero homotopy group pi= in dimension n is called a K(pi,n) space. When n =3D1, pi can be any grou= p. If n > 1, pi has to be commutative. In any case, the (co)homology grou= ps depend only on pi, not on the space. Saunders was working on group extension theory=E2=80=94what we now call exa= ct sequences 0 ---> A --> Pi --> pi --> 1 where A is abelian. He came to A= nn Arbor to give a talk on his work and Sammy immediately recognized that t= hey were making the same kinds of computation. In fact they had both disco= vered H^2(pi,A), which eventually got to be called the second Eilenberg-Mac= Lane cohomology group of pi with coefficients in A. (I'm omitting lots of= details here.) Anyway, they had a long discussion about this. How could = the same computation arise in algebraic topology and (what eventually got t= o be called) homological algebra. In order to start explaining this they discovered the idea of a natural tra= nsformation, for which they needed functors, for which they needed categori= es. The following year, Sammy went to Indiana U. where he met, among others, Cl= ifford Truesdell, the finest 19th century physicist of the 20th century, wh= ich had an interesting consequence, see below. Meantime, the war had started. Saunders moved from being a junior fellow a= t Harvard to a war office in New York. I'm not sure what they were doing t= here, but I would speculate they were creating ballistic firing tables. Wh= ere to aim a cannon given muzzle speed and wind velocity. But he somehow a= rranged to have Sammy join the office. Then every night after work, Sammy = went to Saunders's apartment and they worked. On categories, on the Eilenb= erg-Mac Lane cohomology theory and on the the (co)homology of K(pi,n) space= s. The last was doubtless their deepest work. Or any rate, the one I don'= t really understand. Then the war ended. Sammy stayed in NY, spending the rest of his career at= U. Chicago. They spent five years publishing their work from the war year= s and then their collaboration ceased. Almost surely because they were no = longer in the same location. Mail was slow and unsatisfactory and there wa= s no internet. When Charles and I were trying to collaborate on TTT, mail = between Montreal and Cleveland took a minimum of two weeks. Then we both g= ot computers and we discovered data transfers and we were off. Of course, = collaboration by mail was possible, but highly unsatisfactory. I'm not sure when Bill Lawvere college. Best guess would 1955. He went to= Indiana U. and totally impressed Truesdell. In fact, I have heard that he= ended up living at the Truesdell's. At that time, he was just as interest= ed in physics as in math. But Truesdell felt that Bill's true calling was = math. Although he maintained and interest in physics all his life. At any= rate (and I heard this story from Truesdell himself) when he started think= ing where Bill should go to study math, he recalled the mathematician who h= ad impressed him the most=E2=80=94Sammy. And that explains how Bill ended = up doing his graduate work at Columbia. 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 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 --0000000000008643d2060e2f2c45 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
Isn't this idea of reconstructing G from G/N-->Aut(N)/Out(N) an ear= ly anticipation of Yoneda methods? Amazing. Or am I projecting? In any case= , thank you for a beautiful story beautifully told!

But don't you think that Turing would disagree with the first sentence= of your article? Chatbots will need a while before they can catch up with = being as much of a threat to the health of anything as people are.

Best wishes,
-- dusko

On Mon, Jan 1, 2024 at 1:13=E2=80=AFP= M Johannes Huebschmann <johannes.huebschmann@univ-lille.fr> wrote:
Dear All

Among many items, the EML collaboration developed,
in terms of a 3-cohomology class,
the obstruction to solving the extension problem
for non-abelian groups.

This obstruction is implicit in a paper by A. Turing
published nine  years before EML's paper.

Details are in


Best wishes for the new year.

Johannes



De: "Joyal, Andr=C3=A9" <joyal.andre@uqam.ca>
=C3=80: "Michael Barr, Prof." <barr.michael@mcgill.ca>, "c= ategories" <categories@mq.edu.au>
Envoy=C3=A9: Mardi 26 D=C3=A9cembre 2023 00:56:28
Objet: Re: History: The Eilenberg-Mac Lane collaboration

Dear Michae= l,

Thank you f= or your recollection of collaboration between Sammy and Saunders.

You wrote:<= /span>

  They spent five years publishing their w= ork from the war years and then their collaboration ceased.  Almost surely because they were no longer in the same locati= on.  Mail was slow and unsatisfactory and there was no internet. = When Charles and I were trying to collaborate on TTT, mail between Montrea= l and Cleveland took a minimum of two weeks.  Then we both got computers and we discovered data transfers and we were off.&nb= sp; Of course, collaboration by mail was possible, but highly unsatisfactor= y.

I do not have a direct knowledge of this, but Myles Tierney = told me another story a few decades ago.
As you know, Steenrod discovered his (mod 2) cohomology oper= ations in 1947. 
Eilenberg and MacLane saw that the operations could be expla= ined in terms of the mod 2 cohomology 
of the Eilenberg-MacLane spaces K(Z/2, n).
They decided to compute the cohomology of 
K(A,n) for any abelian group A  and any n >0. &= nbsp;
They published three notes on the cohomology theory of abeli= an groups
 in the Proceedings of the National Academy of Sciences= (1950).
Followed by three papers on the group H(pi,n)  in = 1953.
Cartan was reading their work closely, and he began to work = on
the problem with Serre  in the early 50.
Eilenberg and MacLane had been trying to solve the general p= roblem
by iteratin= g some general contruction (the famous bar resolution).
In contrast, Cartan and Serre concentrated their effort on t= he 
special case of K(Z/p,n) for a prime p.
Also, they exploited systematically the fact that K(A, n) is= the base space of a fibration 
with a cont= ractible total space E(A,n) and fiber K(A,n-1).
Eilenberg e= vetually understood that Cartan's approach was more powerful
and after v= isiting him in Paris 1952 he started a collaboration.
I guess tha= t their Homological Algebra is one upshot.
According t= o Myles, MacLane was upset and he never worked with Eilenberg again.=
He authored= a book on homological algebra using abelian categories years later.=
Fortunately= , Eilenberg and MacLane stayed in contact: 
 they = were the founding fathers of the growing school of category theory!<= /div>

The integra= l homology of K(pi,n) was computed in general by G.J. Decker
in his Phd = thesis (1974) with MacLane, but his description is very complicated.=
Some finite= abelian groups are only described with an infinite presentation!
For a bette= r description in special cases, see the paper of 
Larry Breen= , R. Mikhailov and A. Touze (2013).
Larry was p= lanning to describe them all explicitly.
Unfortunate= ly, he died  May 8, 2023 (after having Alzheimer for many years).
He was a ve= ry good mathematician.
Advancing research. Creating connections.

Best wishes= ,
Andr=C3=A9<= /span>






De :&= nbsp;Michael Barr, Prof. <barr.michael@mcgill.ca>
Envoy=C3=A9 : 17 d=C3=A9cembre 2023 21:19
=C3=80 : categories@mq.edu.au <categories@mq.edu.au>
Objet : History: The Eilenberg-Mac Lane collaboration
 
I would like to tell some of the story of Eilenberg-Mac Lane.  I will = start with a tale told me my a graduate student at Columbia when I began as= an instructor in 1962.

He said that Sammy had never met Saunders in person until the International= Congress in Cambridge, Mass in 1950 and that after they met, they never co= llaborated again. 

It is hard to imagine a more wrong story, but the student appeared to belie= ve it.

In fact, Sammy arrived in the US in (I could be off by a year) 1939 and got= a one year appointment at U. Michigan.  At that time he was working o= n the (co)homology of K(pi,1).  A space with only one non-zero homotop= y group pi in dimension n is called a K(pi,n) space.  When n =3D1, pi can be any group.  If n > 1, pi has t= o be commutative.  In any case, the (co)homology groups depend only on= pi, not on the space.

Saunders was working on group extension theory=E2=80=94what we now call exa= ct sequences 0 ---> A --> Pi --> pi --> 1 where A is abelian.&n= bsp; He came to Ann Arbor to give a talk on his work and Sammy immediately = recognized that they were making the same kinds of computation.  In fact they had both discovered H^2(pi,A), which eventually got to be cal= led the second Eilenberg-Mac Lane cohomology group of pi with coefficients = in A.  (I'm omitting lots of details here.)  Anyway, they had a l= ong discussion about this.  How could the same computation arise in algebraic topology and (what eventually got to b= e called) homological algebra. 

In order to start explaining this they discovered the idea of = a natural transformation, for which they needed functors, for which they needed categories. 

The following year, Sammy went to Indiana U. where he met, amo= ng others, Clifford Truesdell, the finest 19th century physicist of the 20th century= , which had an interesting consequence, see below.

Meantime, the war had started.  Saunders moved from being= a junior fellow at Harvard to a war office in New York.  I'm not sure what they were doing there, but I would sp= eculate they were creating ballistic firing tables.  Where to aim a ca= nnon given muzzle speed and wind velocity.  But he somehow arranged to= have Sammy join the office.  Then every night after work, Sammy went to Saunders's apartment and they worked.  On c= ategories, on the Eilenberg-Mac Lane cohomology theory and on the the (co)h= omology of K(pi,n) spaces.  The last was doubtless their deepest work.=   Or any rate, the one I don't really understand.

Then the war ended.  Sammy stayed in NY, spending the res= t of his career at U. Chicago.  They spent five years publishing their work from the war years and then their collabo= ration ceased.  Almost surely because they were no longer in the same = location.  Mail was slow and unsatisfactory and there was no internet.=   When Charles and I were trying to collaborate on TTT, mail between Montreal and Cleveland took a minimum of two weeks.&n= bsp; Then we both got computers and we discovered data transfers and we wer= e off.  Of course, collaboration by mail was possible, but highly unsa= tisfactory.

I'm not sure when Bill Lawvere college.  Best guess would= 1955.  He went to Indiana U. and totally impressed Truesdell.  In fact, I have heard that he ended up living a= t the Truesdell's.  At that time, he was just as interested in physics= as in math.  But Truesdell felt that Bill's true calling was math.&nb= sp; Although he maintained and interest in physics all his life.  At any rate (and I heard this story from Truesdell himself= ) when he started thinking where Bill should go to study math, he recalled = the mathematician who had impressed him the most=E2=80=94Sammy.  And t= hat explains how Bill ended up doing his graduate work at Columbia.

Michael
 
 
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