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