He has tought at Harvard, Cornell and the University of Chicago. Mac Lane's initial research was in logic and in algebraic number theory valuation theory. With Samuel Eilenberg he published fifteen papers on algebraic topology. A number of them involved the initial steps in the cohomology of groups and in other aspects of homological algebra - as well as the discovery of category theory.

His famous and undergraduate textbook Survey of modern algebra , written jointly with G. Birkhoff, has remained in print for over 50 years. Mac Lane is also the author of several other highly successful books. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

Mathematics Algebra. Classics in Mathematics Free Preview. Buy eBook. Buy Softcover. FAQ Policy. Wilder made Michigan a center of topology, bringing in such figures as. The text of this memoir is reprinted with permission from Notices of the American Mathematical Society, Vol. In Sammy came to the Columbia University mathematics department, which he twice chaired and where he remained till his retirement.

In he was named a University professor, the highest faculty distinction that the university confers. Sammy traveled and collaborated widely. For fifteen years he was a member of Bourbaki. His collaboration with Steenrod produced the book Foundations of Algebraic Topology, that with Henri Cartan the book Homological Algebra, both of them epoch-making works. The Eilenberg-Mac Lane collaboration gave birth to category theory, a field that both men nurtured and followed throughout their ensuing careers. Sammy later brought these ideas to bear in a multivolume work on automata theory. A joint work on topology with Eldon Dyer may see posthumous publication soon.

Among his many honors Sammy won the Wolf Prize shared in with Atle Selberg , was awarded several honorary degrees including one from the University of Pennsylvania , and was elected to membership in the National Academy of Sciences of the USA. His fame among certain art collectors overshadows his mathematical reputation.

In a gesture characteristically marked by its generosity and elegance, Sammy in donated much of his collection to the Metropolitan Museum of Art in New York, which. Samuel Eilenberg died in New York on January 30, , after spending two years in a state of precarious health. I would like to write here of the mathematician and especially of the friend that I gradually discovered in the course of a close collaboration that lasted at least five years and that taught me many things. I met Sammy for the first time at the end of December he had come to greet me at LaGuardia Airport in New York, a city buried under snow, where airplanes had been unable either to take off or to land for two days.

This was my first visit to the United States; it was to last five months. Of course, Eilenberg was not unknown to me, because since the end of the war I had begun to be interested in algebraic topology. Notably I had studied the article in the Annals of Mathematics in which Eilenberg set forth his theory of singular homology one of those theories which immediately takes on a definitive shape. In fact, that formula amounts to a calculation of the homology groups of the tensor product of two graded differential groups as a function of the homology groups of each of them.

The solution involves. At the time of my first meeting with Sammy, I was quite happy with telling that to him. This was the point of departure for our collaboration, by means of postal mail at first. Then Sammy came to spend the year —51 in Paris. Sammy gave two lectures on spectral sequences. Armand Borel and Jean-Pierre Serre took an active part in this seminar also.

We went from discovery to discovery, Sammy having an extraordinary gift for formulating at each moment the conclusions that would emerge from the discussion. And it was always he who wrote everything up as we went along in precise and concise English. After the notion of satellites of a functor came that of derived functors, with their axiomatic characterization.

Gradually the theory included several existing theories cohomology of groups, cohomology of Lie algebras, in the sense of Chevalley and Eilenberg, cohomology of associative algebras. Then came the concept of hyperhomology. Of course, this work together took several years. Sammy made several trips to my country houses in Die and in Dolomieu. Outside of our work hours he participated in our family life.

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Sammy knew how to put his friends to work. I think I remember that he persuaded Steenrod to contribute the preface of our book, where the evolution of the ideas is explained perfectly. He arranged also for other colleagues to collaborate in the writing of the chapter devoted to finite. Our initial project of a mere article for a journal was transformed; it became a book that we would propose to a publisher and for which it would be necessary to find a title that captured its content.

We finally agreed on the term Homological Algebra. The text was given to Princeton University Press in I do not know why the book appeared only in For fifteen years Sammy was also an active member of the Bourbaki group.

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It is therefore very natural that Eilenberg was invited to the Congress that Bourbaki held in October It is necessary to say that he mastered the French language perfectly, which he had learned when he was living in his native Poland. The collaboration of Sammy with Bourbaki lasted until He took part in the summer meetings, which lasted two weeks. He knew admirably how to present his point of view, and he often made us agree to it.

The list made in of his publications comprises, besides 4 books, articles; the first 37 articles are before his emigration from Poland to the United States in , and almost all are written in French. He was not yet twenty years old when he began to publish. The celebrated articles written with S. Mac Lane extended from to The list of his other collaborators is long: N. Steenrod, J. Zilber, T. Nakayama, T.

Ganea, J. Moore, G. Kelly, to cite only the main ones. Starting in , Sammy became actively interested in the theory of automata, which led him to write a book entitled Automata,. I have not mentioned a magnificent collection of sculptures in bronze, silver, or stone, patiently collected in India, Pakistan, Indonesia, Cambodia,…, some of which dated to the third century B. In he gave a great part of his collection to the Metropolitan Museum in New York. In Eilenberg retired from Columbia University, where he had taught since In his mathematical work was recognized by the award of the Wolf Prize in Mathematics, which he shared with Atle Selberg.

Our meeting there was not without emotion. He was for me a friend whose kindness, humor, and faithfulness cannot be forgotten. Samuel Eilenberg, who made decisive contributions to topology and other areas of mathematics, died on Friday, January 30, , in New York City. He had been a leading member of the department of mathematics at Columbia University since His mathematical books, ideas and papers had a major influence. Eilenberg was born in Poland in At the University of Warsaw he was a student of Borsuk in the active school of Polish topology.

His thesis, concerned with the topology of the plane, was published in Fundamenta Mathematica in Its results were well received in Poland and in the. In he published in the same journal another influential paper on the action of the fundamental group on the higher homotopy groups of a space.

Algebra was not foreign to his topology! Get out. At that university Oswald Veblen and Solomon Lefschetz efficiently welcomed refugee mathematicians and found them suitable positions at American universities.

Sammy immediately fitted in, did collaborative research for example, with Wilder, O. Harrold, and Deane Montgomery. He also argued with Lefschetz. Finding the Lefschetz book obscure in its treatment of singular homology, he provided an elegant and definitive treatment in the Annals This I learned when I lectured at Ann Arbor about group extensions. I had calculated an example of group extensions for an interesting factor group involving a prime number p.

## Saunders Mac Lane - Wikiwand

When I told Sammy this result, he immediately saw that it answered a question of Steenrod about the regular cycles of the p -adic solenoid inside a solid torus, wrap another one p times around, and so on, ad infinitum. So Sammy and I stayed up all night to find out the reason for this unexpected appearance of group extensions. Thus Sammy insisted on understanding this unexpected connection between algebra and topology. There was more there: the connection involved mapping topology into algebra, so we were forced to invent functors, natural transformations, and categories to describe this.

All told, this led to our fifteen joint papers. They all involved the maxim: Dig deeper and find out. For example, Hurewicz and Heinz Hopf had observed that the fundamental group of a space had effects on the higher homology and cohomology groups. Sammy, with his knowledge of his singular homology theory, had just the needed tools to understand this, which resulted in our discovery of the cohomology of groups.

Sammy saw that this idea went further, so he started Gerhard Hochschild on his study of the cohomology of algebras and then went on to write, with Henri Cartan, that very influential book on homological algebra, which caught the interest of many algebraists and provided the first book presentation of the important French technique of spectral sequences.

Sammy applied his maxim in other connections. With Joe Zilber he developed the category of simplicial sets as a new type of space—using his singular simplices with face and degeneration operations. With Calvin Elgot he wrote about recursion, a topic in logic. By himself he wrote two volumes on Automata, Languages, and Machines. At that time there were many different and confusing versions of homology theory, some singular, some cellular. This book used categories to show. At Columbia University Sammy took vigorous steps to build up the department. He trained many graduate students.

He was an inspiring teacher. Early in Sammy was felled by a stroke. It became hard for him to talk. In May I was able to visit him; he was lively and passed on to me a not clearly understood proposal. He was then able to spend some time in his apartment on Riverside Drive. I think his message then to me was the same maxim: Keep on pressing those mathematical ideas. This is well illustrated by his life. His ideas—singular homology, categories, simplicial sets, generic acyclicity, obstructions, automata, and the rest—will live on.

The startling idea that homology theory for topological spaces could be used for algebraic objects first arose with the discovery of the cohomology groups of a group. Hurewicz had considered spaces which are aspherical any image of a higher-dimensional sphere can be deformed into a point and had shown that the fundamental group p 1 determines the homotopy type of the space—and hence its homology and cohomology groups. Hopf had then found explicit formulas for the ho-. In other words, old functors lead to new ones. Eilenberg very quickly saw that such cohomological methods would apply to any algebraic situation.

He explained this in the paper [ 2 ]. In each of these cases the cohomology groups in question were the derived functors of naturally occurring Hom functors. These formulas originally were stated to give the Betti numbers and torsion coefficients of a product of two spaces X and Y. This really involved the tensor product of homology groups, and in the famous Eilenberg-Steenrod book it appears in the following short exact sequence:. Also, Tor A,B is a functor of abelian groups, as is ; in fact, Tor turns out to be the first derived functor of!

The definitions of these terms do suffice for the topological task in question: elements of finite order in the groups A and B give elements in Tor. As stated, Tor is the first derived functor of ; it turns out for modules that there are also higher derived functors Tor n A, B for each n. The construction of these higher torsion products and their description by generators and relations were examined by Eilenberg-Mac Lane; these products provided new examples of higher derived functors of modules.

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:. It turns our that the functor Ext A, — is the first derived functor of Hom A, — and thus that there are higher derived functors Ext n A, —.

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They vanish for abelian groups A, but not generally for modules. This provided the background for the influential Cartan-Eilenberg book [ 1 ] on homological algebra. One simply applies the functor to the resolution with the M term dropped and then takes the homology or cohomology of the resulting complex. The ideas of homological algebra were presented in two pioneering books by Cartan-Eilenberg [ 1 ] and Mac Lane [ 4 ].

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The Cartan-Eilenberg treatise had a widespread and decisive influence in algebra. This again illustrates the genius of Eilenberg: If essentially the same idea crops up in different places, follow it out and find out where it lives. When I met Samuel Eilenberg in , he was introduced as Sammy.

He was always referred to as Sammy. It would be wrong to speak of him otherwise. I was then a student; I promptly became his student. I would like to record what drew me then to Sammy and continued over the years to do so—namely, what I perceived as his radical insistence on lucidity, order, and understanding as opposed to trophy hunting, and his idea of how that understanding was to be achieved.

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Perhaps I should illustrate this by a partial in both senses account of his mathematical career. At the end of the s algebraic topology had amassed a stock of problems which its then available tools were unable to attack. Sammy was prominent among a small group of mathematicians—among them, for example, J.

Their success in doing this was attested to by the fact that by the end of the s most of those problems had been solved inordinately many of them by J. With Mac Lane he developed the theory of cohomology of groups, thus providing a proper setting for the remarkable theorem of Hopf on the homology of highly connected spaces. This led them to the study of the Eilenberg-Mac Lane spaces and thus to a deeper understanding of the relations between homotopy and homology. Their most fateful invention perhaps was that of category.

His e-mail address is aheller email. In collaboration with Steenrod, Sammy drained the Pontine Marshes of homology theory, turning an ugly morass of variously motivated constructions into a simple and elegant system of axioms applied, for the first time, to functors. This was a radical innovation. Heretofore homology theories had been procedures for computing; henceforth they would be mathematical objects in their own right.

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What was especially remarkable was that in order to achieve this, Sammy and Steenrod undertook to raise the logical level of the things that might be so regarded. The algebraic structures of the new algebraic topology were proving themselves useful in other parts of mathematics: in algebra, representation theory, algebraic geometry, and even in number theory. Together with Henri Cartan, Sammy systematized these structures under the rubric of Homological Algebra, once more raising the level of discourse by introducing such notions as derived functors.

I am tempted to insert a parenthesis here.

Since homological algebra has proved indispensable, the honors lie, I think, with Cartan and Eilenberg. The roots of homological algebra lay nevertheless in algebraic topology, and Sammy, in collaboration with John Moore, returned to these. They introduced such novelties as differential graded homological algebra and relative homological algebra to provide homes for the new techniques intro-. Notable among them are the so-called Eilenberg-Moore spectral sequences, which deal with pullbacks of fibrations and with associated fiber bundles.

Unfortunately neither Sammy nor his last collaborator, Eldon Dyer, lived to complete their ultimate project of refounding algebraic topology in the correct—which is to say, homotopical—setting. Perhaps this project was too ambitious. I learned from Eldon how much agony accompanied even such choices as that of the correct definition of a topological space.

Some part of their book may yet survive, and others are already continuing their project piecemeal. As I perceived it, then, Sammy considered that the highest value in mathematics was to be found, not in specious depth nor in the overcoming of overwhelming difficulty, but rather in providing the definitive clarity that would illuminate its underlying order. This was to be accomplished by elucidating the true structure of the objects of mathematics.

Let me hasten to say that this was in no sense an ontological quest: the true structure was intrinsic to mathematics and was to be discerned only by doing more mathematics. Sammy had no patience for metaphysical argument. He was not a Platonist; equally, he was not a non-Platonist. Category theory also developed into a mathematical subject with its own honorable history and practitioners, beginning with Mac Lane and including, notably, F. Sammy did not, I think, want to be reckoned a member of this school.

I believe, in fact, that he would have rejected the idea that mathematics needed a foundation.