Biographical Memoirs Volume 79 (2001) / Chapter Skim
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Samuel Eilenberg
Samuel Eilenberg
Pages 106-133
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From page 107... ...
The ideas that accomplished this were so fundamental and supple that they took on a life of their own, giving birth first to homological algebra en cl in turn to category theory, structures that now permeate much of contemporary mathematics. Born in Warsaw, Poland, Sammy stucliec!
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From page 108... ...
His collaboration with Steenrocl proclucecl the book Foundations of Algebraic Topology, that with Henri Cartan the book Homolog~cal Algebra, both of them epoch-making works. The Eilenberg-Mac Lane collaboration gave birth to category theory, a fielcl that both men nurturer!
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From page 109... ...
on the "Kunneth formula", which gives the Betti numbers en cl the torsion coefficients of the product of two simplicial complexes. In fact, that formula amounts to a calculation of the homology groups of the tensor product of two graclecl clifferential groups as a function of the homology groups of each of them.
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From page 110... ...
concise English. After the notion of satellites of a functor came that of derived functors, with their axiomatic characterization.
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From page 111... ...
The above gives only a faint iclea of Samuel Eilenberg's mathematical activity. The list macle in 1974 of his publications comprises, besicles 4 books, Ill articles, the first 37 articles are before his emigration from Polancl to the Unitecl States in 1939, en cl almost all are written in French.
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From page 112... ...
Its results were well received in Poland and in the Saunders Mac Lane is Maz Mason Distinguished Service Professor, Emeritus, at the University of Chicago.
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From page 113... ...
Get out." He clicI, arriving in New York on April 23,1939, and going at once to Princeton. At that university Oswalcl Veblen en cl Solomon Lefschetz efficiently welcomecl refugee mathematicians and found them suitable positions at American universities.
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From page 114... ...
HochschiTc! on his stucly of the cohomology of algebras en cl then went on to write, with Henri Cartan, that very influential book on homological algebra, which caught the interest of many algebraists en c!
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From page 115... ...
For example, his students en cl postclocs in category theory incluclecl Harry Applegate, Mike Barr, Jonathan Beck, David Buchsbaum, Peter FreycI, Alex Heller, Daniel Kan, Bill Lawvere, Frecl Linton, Steve Schanuel, Myles Tierney, en cl others. He was an inspiring teacher.
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From page 116... ...
~ This really involved the tensor product of homology groups, en cl in the famous Eilenberg-Steenrocl book it appears in the following short exact sequence: ,
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From page 117... ...
Now return to the functor Ext(A, B) , the group of abelian group extensions E of B by A, so that E appears in a short exact sequence of abelian groups: O ~ B ~ E ~ A ~ O
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From page 118... ...
the computation from specific "bar resolutions" usecl to define the cohomology of a group. The icleas of homological algebra were presented in two pioneering books by Cartan-Eilenberg ~ ~ ~ and Mac Lane t4]
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From page 119... ...
C WhiteheacI, Hassler Whitney, Saunders Mac Lane, en c!
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From page 120... ...
This latest innovation brought its authors into conflict with the "establishment" by putting in question the very notion of definition, raising a fundamental question of the relation between category theory and set theory that has yet to be put definitively to rest. Since homological algebra has proved indispensable, the honors lie, I think, with Cartan and Eilenberg.
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From page 121... ...
Category theory also clevelopecl into a mathematical subject with its own honorable history en c! practitioners, beginning with Mac Lane and including, notably, F
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From page 122... ...
PETER FREYD Thirty years ago I fount! myself a neighbor of Arthur Upham Pope, the master of ancient Persian art.
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From page 123... ...
art, at something of a distance. But both worlds seemed to agree on a one thing, the very one that Arthur Upham Pope had insisted upon: Sammy was the clearer.
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From page 124... ...
no more aggressive than, say, the sun rising at its appointed sunrise time. Forty years ago Sammy hopecl to turn the study of Indian bronzes into an equally well-behavec!
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From page 125... ...
" Style is only one part of his mathematics as, of course, he knew but there are, incleecI, wonclerful stories about Sammy, attending only to what seemed the most superficial of stylistic choices, restructuring entire subjects on the spot. Many have witnessed this triumph of style over substance, particularly with students.
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From page 126... ...
Sammy's view of Poland since the war was more complicatecI. It was particularly complicatecl by what he viewocl as its treatment of category theory as a fringe subject.
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From page 127... ...
He hacl watchecl many of his inventions become stanciarc! mathematicssingular homology, obstruction theory, homological algebra en cl he hacl no intention of leaving the future of category theory to others.
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From page 128... ...
For the next thirty-five years he went to just about every category theory conference, and, much more important, he usecl his masterly expository skills to convey categorical icleas to other mathematicians. Sammy's efforts succeeclec!
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From page 129... ...
Research in topology, algebraic geometry, complex analysis, number theory, en cl the then budding category theory were quite active there. Though a faculty member, I functioned!
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From page 130... ...
His was a most satisfying and inspiring influence on my own professional life. After his stroke, it was painful to see Sammy, frail en cl gaunt en cl cleprivecl of speech when his still active mind had so much yet to say.
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From page 131... ...
S EILENBERG, Topological methods in abstract algebra: Cohomology theory of groups, Bull.
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From page 132... ...
, 1 37-1 46. Singular homology theory, Ann.
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From page 133... ...
, 43-53. With Cartan, E., Homological Algebra, Princeton, Univ.
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