*History: The Eilenberg-Mac Lane collaboration@ 2023-12-18 2:19 Michael Barr, Prof.2023-12-25 23:56 ` Joyal, André 0 siblings, 1 reply; 5+ messages in thread From: Michael Barr, Prof. @ 2023-12-18 2:19 UTC (permalink / raw) To: categories [-- Attachment #1: Type: text/plain, Size: 4302 bytes --] 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 collaborated again. It is hard to imagine a more wrong story, but the student appeared to believe 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 =1, pi can be any group. If n > 1, pi has to be commutative. In any case, the (co)homology groups depend only on pi, not on the space. Saunders was working on group extension theory—what we now call exact sequences 0 ---> A --> Pi --> pi --> 1 where A is abelian. 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 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 to be 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, among 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 speculate they were creating ballistic firing tables. Where to aim a cannon 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 categories, on the Eilenberg-Mac Lane cohomology theory and on the the (co)homology 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 rest of his career at U. Chicago. They spent five years publishing their work from the war years and then their collaboration 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. Then we both got 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 interested 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 thinking where Bill should go to study math, he recalled the mathematician who had impressed him the most—Sammy. 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 mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g> [-- Attachment #2: Type: text/html, Size: 10499 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: History: The Eilenberg-Mac Lane collaboration2023-12-18 2:19 History: The Eilenberg-Mac Lane collaboration Michael Barr, Prof.@ 2023-12-25 23:56 ` Joyal, André2024-01-01 12:20 ` Johannes Huebschmann 0 siblings, 1 reply; 5+ messages in thread From: Joyal, André @ 2023-12-25 23:56 UTC (permalink / raw) To: Michael Barr, Prof., categories [-- Attachment #1: Type: text/plain, Size: 7717 bytes --] 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 their collaboration 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. Then we both got computers and we discovered data transfers and we were off. Of course, collaboration 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 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. 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 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! 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 complicated. 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). <https://arxiv.org/pdf/1312.5676<https://protect-au.mimecast.com/s/MIitCzvkmpfAlX2xhXn7W3?domain=arxiv.org>> 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=7249<https://protect-au.mimecast.com/s/wRNuCANpnDCzyxWnf9lSG6?domain=ams.org> News from the AMS<https://protect-au.mimecast.com/s/wRNuCANpnDCzyxWnf9lSG6?domain=ams.org> Advancing research. Creating connections. www.ams.org<https://protect-au.mimecast.com/s/eRdjCBNqgBCOQx4MhjrfJK?domain=ams.org> Best wishes, André ________________________________ De : Michael Barr, Prof. <barr.michael@mcgill.ca> Envoyé : 17 décembre 2023 21:19 À : 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 collaborated again. It is hard to imagine a more wrong story, but the student appeared to believe 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 =1, pi can be any group. If n > 1, pi has to be commutative. In any case, the (co)homology groups depend only on pi, not on the space. Saunders was working on group extension theory—what we now call exact sequences 0 ---> A --> Pi --> pi --> 1 where A is abelian. 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 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 to be 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, among 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 speculate they were creating ballistic firing tables. Where to aim a cannon 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 categories, on the Eilenberg-Mac Lane cohomology theory and on the the (co)homology 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 rest of his career at U. Chicago. They spent five years publishing their work from the war years and then their collaboration 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. Then we both got 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 interested 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 thinking where Bill should go to study math, he recalled the mathematician who had impressed him the most—Sammy. 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 mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g> [-- Attachment #2: Type: text/html, Size: 26134 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: History: The Eilenberg-Mac Lane collaboration2023-12-25 23:56 ` Joyal, André@ 2024-01-01 12:20 ` Johannes Huebschmann2024-01-05 9:07 ` Dusko Pavlovic 0 siblings, 1 reply; 5+ messages in thread From: Johannes Huebschmann @ 2024-01-01 12:20 UTC (permalink / raw) To: Joyal, André;+Cc:Michael Barr, Prof., categories [-- Attachment #1: Type: text/plain, Size: 8472 bytes --] 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<https://protect-au.mimecast.com/s/C6k4CL7Eg9fZz5RzUBz2vd?domain=ams.org> Best wishes for the new year. Johannes ________________________________ De: "Joyal, André" <joyal.andre@uqam.ca> À: "Michael Barr, Prof." <barr.michael@mcgill.ca>, "categories" <categories@mq.edu.au> Envoyé: Mardi 26 Décembre 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 their collaboration 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. Then we both got computers and we discovered data transfers and we were off. Of course, collaboration 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 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. 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 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! 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 complicated. 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). <https://arxiv.org/pdf/1312.5676<https://protect-au.mimecast.com/s/1PlICMwGj8CAnGqnHk-4AK?domain=arxiv.org>> 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=7249<https://protect-au.mimecast.com/s/aYsLCOMK7YcyBopBtrjIsS?domain=ams.org> News from the AMS<https://protect-au.mimecast.com/s/aYsLCOMK7YcyBopBtrjIsS?domain=ams.org> Advancing research. Creating connections. www.ams.org<https://protect-au.mimecast.com/s/aIVSCP7L1NfkLqKLH6RQ6W?domain=ams.org> Best wishes, André ________________________________ De : Michael Barr, Prof. <barr.michael@mcgill.ca> Envoyé : 17 décembre 2023 21:19 À : 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 collaborated again. It is hard to imagine a more wrong story, but the student appeared to believe 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 =1, pi can be any group. If n > 1, pi has to be commutative. In any case, the (co)homology groups depend only on pi, not on the space. Saunders was working on group extension theory—what we now call exact sequences 0 ---> A --> Pi --> pi --> 1 where A is abelian. 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 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 to be 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, among 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 speculate they were creating ballistic firing tables. Where to aim a cannon 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 categories, on the Eilenberg-Mac Lane cohomology theory and on the the (co)homology 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 rest of his career at U. Chicago. They spent five years publishing their work from the war years and then their collaboration 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. Then we both got 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 interested 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 thinking where Bill should go to study math, he recalled the mathematician who had impressed him the most—Sammy. 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 mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g> [-- Attachment #2: Type: text/html, Size: 28156 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: History: The Eilenberg-Mac Lane collaboration2024-01-01 12:20 ` Johannes Huebschmann@ 2024-01-05 9:07 ` Dusko Pavlovic2024-01-05 19:43 ` Dusko Pavlovic 0 siblings, 1 reply; 5+ messages in thread From: Dusko Pavlovic @ 2024-01-05 9:07 UTC (permalink / raw) To: Johannes Huebschmann;+Cc:Joyal, André, Michael Barr, Prof., categories [-- Attachment #1: Type: text/plain, Size: 9744 bytes --] Isn't this idea of reconstructing G from G/N-->Aut(N)/Out(N) an early 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 PM Johannes Huebschmann <johannes.huebschmann@univ-lille.fr<mailto: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 https://www.ams.org/journals/notices/202311/rnoti-p1802.pdf<https://protect-au.mimecast.com/s/QARnCr810kCv8B1vt7h_Ek?domain=ams.org> Best wishes for the new year. Johannes ________________________________ De: "Joyal, André" <joyal.andre@uqam.ca<mailto:joyal.andre@uqam.ca>> À: "Michael Barr, Prof." <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>>, "categories" <categories@mq.edu.au<mailto:categories@mq.edu.au>> Envoyé: Mardi 26 Décembre 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 their collaboration 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. Then we both got computers and we discovered data transfers and we were off. Of course, collaboration 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 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. 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 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! 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 complicated. 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). <https://arxiv.org/pdf/1312.5676<https://protect-au.mimecast.com/s/JnijCwV1jpS1GwA1i9wgta?domain=arxiv.org>> 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=7249<https://protect-au.mimecast.com/s/q-MaCxngGkfA1W9AtwpZW0?domain=ams.org> News from the AMS<https://protect-au.mimecast.com/s/q-MaCxngGkfA1W9AtwpZW0?domain=ams.org> Advancing research. Creating connections. www.ams.org<https://protect-au.mimecast.com/s/RmtwCzvkmpf2Mjx2hwT8Jz?domain=ams.org> Best wishes, André ________________________________ De : Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> Envoyé : 17 décembre 2023 21:19 À : categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto: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 collaborated again. It is hard to imagine a more wrong story, but the student appeared to believe 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 =1, pi can be any group. If n > 1, pi has to be commutative. In any case, the (co)homology groups depend only on pi, not on the space. Saunders was working on group extension theory—what we now call exact sequences 0 ---> A --> Pi --> pi --> 1 where A is abelian. 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 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 to be 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, among 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 speculate they were creating ballistic firing tables. Where to aim a cannon 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 categories, on the Eilenberg-Mac Lane cohomology theory and on the the (co)homology 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 rest of his career at U. Chicago. They spent five years publishing their work from the war years and then their collaboration 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. Then we both got 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 interested 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 thinking where Bill should go to study math, he recalled the mathematician who had impressed him the most—Sammy. 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 mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://protect-au.mimecast.com/s/gW0iCANpnDCWNDnWS26629?domain=outlook.office365.com> | Leave group<https://protect-au.mimecast.com/s/iP-vCBNqgBC47LM4Urc-1e?domain=outlook.office365.com> | Learn more about Microsoft 365 Groups<https://protect-au.mimecast.com/s/jJVaCD1vRkCV58nVT8Esle?domain=aka.ms> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g> [-- Attachment #2: Type: text/html, Size: 29431 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*2024-01-05 9:07 ` Dusko PavlovicRe: History: The Eilenberg-Mac Lane collaboration@ 2024-01-05 19:43 ` Dusko Pavlovic0 siblings, 0 replies; 5+ messages in thread From: Dusko Pavlovic @ 2024-01-05 19:43 UTC (permalink / raw) To: Johannes Huebschmann;+Cc:Joyal, André, Michael Barr, Prof., categories [-- Attachment #1: Type: text/plain, Size: 777 bytes --] sorry, instead Isn't this idea of reconstructing G from G/N-->Aut(N)/Out(N) i should have written "reconstructing G from G/N-->Out(N)=Aut(N)/Inn(N)" You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g> [-- Attachment #2: Type: text/html, Size: 3951 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

end of thread, other threads:[~2024-01-05 19:56 UTC | newest]Thread overview:5+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2023-12-18 2:19 History: The Eilenberg-Mac Lane collaboration Michael Barr, Prof. 2023-12-25 23:56 ` Joyal, André 2024-01-01 12:20 ` Johannes Huebschmann 2024-01-05 9:07 ` Dusko Pavlovic 2024-01-05 19:43 ` Dusko Pavlovic

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