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charset="utf-8" Content-Transfer-Encoding: quoted-printable 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 --=_9b3d1d9c-7d59-4489-9787-cb7e5be985f7 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable