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charset="Windows-1252" Content-Transfer-Encoding: quoted-printable 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=97what we now call exact seq= uences 0 ---> A --> Pi --> pi --> 1 where A is abelian. He came to Ann Arb= or to give a talk on his work and Sammy immediately recognized that they we= re 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 detai= ls here.) Anyway, they had a long discussion about this. How could the sa= me computation arise in algebraic topology and (what eventually got to be c= alled) 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=97Sammy. And that explains how Bill ended up doi= ng 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 --_000_YQXPR01MB3927D239FA7583AE8B821DCC9C90AYQXPR01MB3927CANP_ Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable
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=97what we now call exact seq= uences 0 ---> A --> Pi --> pi --> 1 where A is abelian.  H= e came to Ann Arbor to give a talk on his work and Sammy immediately recogn= ized 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 discovere= d the idea of a natural transformation, for which they needed functors, for which they needed categories. 

The following year, Sammy went to Indiana U. whe= re he met, among others, Clifford Truesdell, the finest 19th century physicist of the 20th&n= bsp;century, which had an interesting consequence, see below.

Meantime, the war had started.  Saunders mo= ved 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.  W= here to aim a cannon given muzzle speed and wind velocity.  But he som= ehow 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 t= he the (co)homology of K(pi,n) spaces.  The last was doubtless their d= eepest work.  Or any rate, the one I don't really understand.

Then the war ended.  Sammy stayed in NY, sp= ending the rest of his career at U. Chicago.  They spent five years publishing their work from the war years and then th= eir collaboration ceased.  Almost surely because they were no longer i= n the same location.  Mail was slow and unsatisfactory and there was n= o 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 trans= fers and we were off.  Of course, collaboration by mail was possible, = but highly unsatisfactory.

I'm not sure when Bill Lawvere college.  Be= st guess would 1955.  He went to Indiana U. and totally impressed Truesdell.  In fact, I have heard that he en= ded up living at the Truesdell's.  At that time, he was just as intere= sted in physics as in math.  But Truesdell felt that Bill's true calli= ng was math.  Although he maintained and interest in physics all his life.  At any rate (and I heard this story from Tr= uesdell himself) when he started thinking where Bill should go to study mat= h, he recalled the mathematician who had impressed him the most=97Sa= mmy.  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   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
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