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Samuel Eilenberg (1913–1998)

Hyman Bass, , Peter Freyd, Alex Heller, and

Samuel Eilenberg died in New York, January 30, Sammy traveled and collaborated widely. For fif- 1998, after a two-year illness brought on by a teen years he was a member of Bourbaki. His col- stroke. He left no surviving family, except for his laboration with Steenrod produced the book Foun- wide family of friends, students, and colleagues, dations of , that with Henri and the rich legacy of his life’s work, in both math- Cartan the book Homological , both of ematics and as an art collector. them epoch-making works. The Eilenberg-Mac Lane “Sammy”, as he has long been called by all who collaboration gave birth to , a field had the good fortune to know him, was one of the that both men nurtured and followed throughout great architects of twentieth-century their ensuing careers. Sammy later brought these and definitively reshaped the ways we think about ideas to bear in a multivolume work on automata topology. The ideas that accomplished this were theory. A joint work on topology with Eldon Dyer so fundamental and supple that they took on a life may see posthumous publication soon. of their own, giving birth first to homological al- Among his many honors Sammy won the Wolf gebra and in turn to category theory, structures that Prize (shared in 1986 with ), was now permeate much of contemporary mathemat- awarded several honorary degrees (including one ics. from the University of Pennsylvania), and was Born in , , Sammy studied in the elected to membership in the National Academy Polish school of topology. At his father’s urging, of Sciences of the USA. On the occasion of the he fled Europe in 1939. On his arrival in Princeton, honorary degree at the University of Pennsylvania and helped him in 1985, he was cited as “our greatest mathemat- (as they had helped other refugees) find a position ical stylist”. at the , where Ray Wilder was The aesthetic principles that guided Sammy’s building up a group in topology. Wilder made mathematical work also found expression in his Michigan a center of topology, bringing in such fig- passion for art collecting. Over the years Sammy ures as , , Hans Samel- gathered one of the world’s most important col- son, and others. Saunders Mac Lane’s invited lec- lections of Southeast Asian art. His fame among ture there on group extensions precipitated the certain art collectors overshadows his mathemat- long and fruitful Eilenberg-Mac Lane collabora- ical reputation. In a gesture characteristically tion. marked by its generosity and elegance, Sammy in In 1947 Sammy came to the Columbia Univer- 1987 donated much of his collection to the Met- sity mathematics department, which he twice ropolitan Museum of Art in New York, which in turn chaired and where he remained till his retire- was thus motivated to contribute substantially to ment. In 1982 he was named a University Pro- the endowment of the Eilenberg Visiting Profes- fessor, the highest faculty distinction that the sorship in Mathematics at . university confers. —

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the sense of Cheval- Henri Cartan ley and Eilenberg, Samuel Eilenberg died in New York on January of as- 30, 1998, after spending two years in a state of pre- sociative ). carious health. I would like to write here of the Then came the con- and especially of the friend that I cept of hyperho- gradually discovered in the course of a close col- mology. laboration that lasted at least five years and that Of course, this taught me many things. work together took I met Sammy for the first time at the end of De- several years. cember 1947: he had come to greet me at La- Sammy made sev- Guardia Airport in New York, a city buried under eral trips to my snow, where airplanes had been unable either to country houses (in take off or to land for two days. This was my first Die and in visit to the ; it was to last five months. Dolomieu). Outside Of course, Eilenberg was not unknown to me, be- of our work hours cause since the end of the war I had begun to be he participated in interested in algebraic topology. Notably I had our family life. studied the article in the 1944 Annals of Math- Sammy knew ematics in which Eilenberg set forth his theory of how to put his singular (one of those theories which im- friends to work. I mediately takes on a definitive shape). I had, for think I remember my part, reflected on the “Künneth formula”, which that he persuaded

gives the Betti numbers and the torsion coeffi- Steenrod to con- Photograph courtesy of Columbia University. tribute the preface cients of the product of two simplicial complexes. Samuel Eilenberg In fact, that formula amounts to a calculation of of our book, where the homology groups of the tensor product of two the evolution of the graded differential groups as a function of the ho- ideas is explained perfectly. He arranged also for mology groups of each of them. The solution in- other colleagues to collaborate in the writing of the volves not only the tensor product of the homol- chapter devoted to finite groups. Our initial pro- ogy groups of the factors but also a new functor ject of a mere article for a journal was transformed; of these groups, the functor Tor. At the time of it became a book that we would propose to a pub- my first meeting with Sammy, I was quite happy lisher and for which it would be necessary to find with telling that to him. a title that captured its content. We finally agreed This was the point of departure for our collab- on the term . The text was oration, by means of postal mail at first. Then given to Press in 1953. I do Sammy came to spend the year 1950–51 in Paris. not know why the book appeared only in 1956. He took part in my seminar at the École Normale, For fifteen years Sammy was also an active devoted that year to cohomology of groups, spec- member of the Bourbaki group. It was, I think, in tral sequences, and sheaf theory. Sammy gave two 1949 that André Weil, who was living in the United lectures on spectral sequences. Armand Borel and States, made contact with him in order to have him Jean-Pierre Serre took an active part in this semi- collaborate on a draft for use by Bourbaki, entitled nar also. “SEAW Report on Homotopy Groups and Fiber Independently of the seminar, Sammy and I had Spaces”. It is therefore very natural that Eilenberg work sessions with the aim of writing an article that was invited to the Congress that Bourbaki held in would develop some of the new ideas born out of October 1950. He was immediately appreciated the Künneth formula. We went from discovery to and became a member of the group under the discovery, Sammy having an extraordinary gift for name “Sammy”. It is necessary to say that he mas- formulating at each moment the conclusions that tered the French language perfectly, which he had would emerge from the discussion. And it was al- learned when he was living in his native Poland. ways he who wrote everything up as we went along The collaboration of Sammy with Bourbaki in precise and concise English. After the notion of lasted until 1966. He took part in the summer satellites of a functor came that of derived functors, meetings, which lasted two weeks. He knew ad- with their axiomatic characterization. Gradually the mirably how to present his point of view, and he theory included several existing theories (coho- often made us agree to it. mology of groups, cohomology of Lie algebras, in The above gives only a faint idea of Samuel Henri Cartan is professor emeritus of mathematics at Eilenberg’s mathematical activity. The list made in Université de Paris XI. This segment is translated and 1974 of his publications comprises, besides 4 adapted from the Gazette des Mathématiciens by per- books, 111 articles; the first 37 articles are before mission. his emigration from Poland to the United States in

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1939, and almost all are written in French. He was position for him was found at the University of not yet twenty years old when he began to publish. Michigan. There Ray Wilder had an active group of The celebrated articles written with S. Mac Lane ex- topologists, including Norman Steenrod, then a tended from 1942 to 1954. The list of his other col- recent Princeton Ph.D. Sammy immediately fitted laborators is long: N. E. Steenrod, J. A. Zilber, in, did collaborative research (for example, with T. Nakayama, T. Ganea, J. C. Moore, G. M. Kelly, to Wilder, O. G. Harrold, and ). cite only the main ones. Starting in 1966, Sammy His 1940 paper in the for- became actively interested in the theory of au- mulated and codified the ideas of the “obstruc- tomata, which led him to write a book entitled Au- tions” recently introduced by . He tomata, Languages, and Machines, published in also argued with Lefschetz. Finding the Lefschetz 1974 by Academic Press. book (1942) obscure in its treatment of singular I have not mentioned a magnificent collection homology, he provided an elegant and definitive of sculptures in bronze, silver, or stone, patiently treatment in the Annals (1944). collected in India, Pakistan, Indonesia, Cambo- Sammy’s idea was to dig deep and deeper till dia,…, some of which dated to the third century he got to the bottom of each issue. This I learned B.C. In 1967 he gave a great part of his collection when I lectured at Ann Arbor about group exten- to the Metropolitan Museum in New York. sions. I had calculated an example of the group of In 1982 Eilenberg retired from Columbia Uni- group extensions for an interesting factor group versity, where he had taught since 1947. In 1986 involving a prime number p. When I told Sammy his mathematical work was recognized by the this result, he immediately saw that it answered a award of the Wolf Prize in Mathematics, which he question of Steenrod about the regular cycles of shared with Atle Selberg. the p-adic solenoid (inside a solid torus, wrap an- The last time I saw Sammy was when the Uni- other one p times around, and so on, ad infinitum). versité de Louvain-la-Neuve organized a conference So Sammy and I stayed up all night to find out the in his honor. Our meeting there was not without reason for this unexpected appearance of group emotion. He was for me a friend whose kindness, extensions. We found out more: it rested on a “uni- humor, and faithfulness cannot be forgotten. versal coefficient theorem” which gave cohomol- ogy with any coefficient group G in terms of ho- mology and an exact sequence involving Ext, the Saunders Mac Lane group of group extensions. Thus Sammy insisted on understanding this unexpected connection be- Samuel Eilenberg, who made decisive contribu- tween algebra and topology. There was more there: tions to topology and other areas of mathematics, the connection involved mapping topology into died on Friday, January 30, 1998, in New York algebra, so we were forced to invent functors, nat- City. He had been a leading member of the de- ural transformations, and categories to describe partment of mathematics at Columbia University this. All told, this led to our fifteen joint papers. since 1947. His mathematical books, ideas, and pa- They all involved the maxim: Dig deeper and find pers had a major influence. out. For example, Hurewicz and Heinz Hopf had Eilenberg was born in Poland in 1913. At the Uni- observed that the fundamental group of a space versity of Warsaw he was a student of Borsuk in had effects on the higher homology and coho- the active school of Polish topology. His thesis, con- mology groups. Sammy, with his knowledge of his cerned with the topology of the plane, was pub- singular homology theory, had just the needed lished in Fundamenta Mathematica in 1936. Its tools to understand this, which resulted in our results were well received both in Poland and in discovery of the cohomology of groups. Sammy saw the USA. In 1938 he published in the same jour- that this idea went further, so he started Gerhard nal another influential paper on the action of the Hochschild on his study of the cohomology of al- fundamental group on the higher homotopy groups gebras and then went on to write, with Henri Car- of a space. Algebra was not foreign to his topol- tan, that very influential book on homological al- ogy! gebra, which caught the interest of many Early in 1939 Sammy’s father told him, “Sammy, algebraists and provided the first book presenta- it doesn’t look good here in Poland. Get out.” He tion of the important French technique of spectral did, arriving in New York on April 23, 1939, and sequences. going at once to Princeton. At that university Os- Sammy applied his maxim in other connections. wald Veblen and Solomon Lefschetz efficiently With Joe Zilber he developed the category of sim- welcomed refugee and found them plicial sets as a new type of space—using his sin- suitable positions at American universities. gular simplices with face and degeneration oper- Sammy’s work in topology was well known, so a ations. With Calvin Elgot he wrote about recursion, Saunders Mac Lane is Max Mason Distinguished Service a topic in logic. By himself he wrote two volumes Professor, Emeritus, at the . on Automata, Languages, and Machines. And with

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Eldon Dyer he prepared two volumes (not yet pub- and taken modulo the “principal” such—those f lished) on General and Categorical Topology. given as f (x)=xa a for some a in A. The ele- n − Algebraic topology was decisively influenced ments of H (π1,A) were functions f (x1,...,xn) of by Eilenberg’s earlier 1952 work with Norman n elements xi satisfying a suitable equation, mod- Steenrod, entitled Foundations of Algebraic Topol- ulo trivial solutions. In other words, the coho- ogy. At that time there were many different and mology of π1 was given as the cohomology of a confusing versions of homology theory, some sin- certain chain complex, the so-called “bar resolu- gular, some cellular. This book used categories to tion”. In the terminology subsequently refined by n st show that they all could be described conceptually Cartan-Eilenberg, H (π1, —) was the (n 1) 1 − as presenting homology functors from the category “derived” functor of H (π1, —). In other words, old of pairs of spaces to groups or to rings, satisfying functors lead to new ones. suitable axioms such as “excision”. Thanks to Eilenberg very quickly saw that such cohomo- Sammy’s insight and his enthusiasm, this text logical methods would apply to any algebraic sit- drastically changed the teaching of topology. uation. He explained this in the 1949 paper [2]. In At Columbia University Sammy took vigorous 1948 he wrote with Chevalley a paper on the co- steps to build up the department. He trained many homology theory of Lie algebras, and about the graduate students. For example, his students and same time he encouraged Gerhard Hochschild, postdocs in category theory included Harry Ap- then one of Chevalley’s Ph.D. students, to introduce plegate, Mike Barr, Jonathan Beck, David Buchs- cohomology groups for associative algebras. In baum, Peter Freyd, Alex Heller, , Bill each of these cases the cohomology groups in Lawvere, Fred Linton, Steve Schanuel, Myles Tier- question were the derived functors of naturally oc- ney, and others. He was an inspiring teacher. curring Hom functors. Classical questions of al- Early in 1996 Sammy was felled by a stroke. It gebraic topology also entered by way of the Kün- became hard for him to talk. In May 1997 I was able neth formulas. These formulas originally were to visit him; he was lively and passed on to me a stated to give the Betti numbers and torsion coef- not clearly understood proposal. He was then able ficients of a product of two spaces X and Y. This to spend some time in his apartment on Riverside really involved the tensor product of homology Drive. I think his message then to me was the same groups, and in the famous Eilenberg-Steenrod book maxim: Keep on pressing those mathematical ideas. it appears in the following short exact sequence: This is well illustrated by his life. His ideas—sin- gular homology, categories, simplicial sets, generic acyclicity, obstructions, automata, and the rest— 0 Hm(X) Hq(Y ) Hn(X Y ) will live on. → ⊗ → × m+Xq=n Our fifteen joint research papers have been col- Tor(Hm(X),Hq(Y )) 0. lected in the volume Eilenberg/Mac Lane, Collected → → m+q=n 1 Works, Academic Press, Inc., New York, 1988. X −

Here “exact” means that at each point the image Next, I comment on Eilenberg’s contributions to of the incoming arrow is the kernel of the outgo- the sources of homological algebra. The startling ing arrow. Also, Tor(A, B) is a functor of abelian idea that homology theory for topological spaces groups, as is ; in fact, Tor turns out to be the first ⊗ could be used for algebraic objects first arose with derived functor of ! The definitions of these ⊗ the discovery of the cohomology groups of a group. terms do suffice for the topological task in ques- Hurewicz had considered spaces which are as- tion: elements of finite order in the groups A and pherical (any image of a higher-dimensional sphere B give elements in Tor. I clearly recall an occasion can be deformed into a point) and had shown that when I tried to explain to Professor Künneth at Er- the fundamental group π determines the homo- 1 langen University that this abstract language did topy type of the space—and hence its homology indeed produce his original numerical Künneth and cohomology groups. Hopf had then found ex- formulas. As stated, Tor is the first derived func- plicit formulas for the homology (Betti) groups of tor of ; it turns out for modules that there are such a space. Then Eilenberg-Mac Lane exhibited ⊗ also higher derived functors Tor (A, B) for each n. the nth cohomology group Hn(X,A) of such a n The construction of these higher torsion products space with coefficients in an abelian group A as and their description by generators and relations a functor of π and A —the nth cohomology 1 were examined by Eilenberg-Mac Lane; these prod- Hn(π ,A) of the group π with coefficients in the 1 1 ucts provided new examples of higher derived π -module A. In particular H1 was simply the 1 functors of modules. For abelian groups A and B, group of “crossed homomorphisms” f : π1 A → Tor (A, B)=0when n>1. satisfying n Now return to the functor Ext(A, B), the group f (xy)=xf (y)+f (x) of abelian group extensions E of B by A, so that

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E appears in a short exact sequence of abelian Perhaps I should illustrate this by a partial (in groups: both senses) account of his mathematical career. 0 B E A 0. At the end of the 1930s algebraic topology had → → → → It turns out that the functor Ext(A, —) is the first amassed a stock of problems which its then avail- derived functor of Hom(A, —) and thus that there able tools were unable to attack. Sammy was promi- are higher derived functors Extn(A, —). They van- nent among a small group of mathematicians— ish for abelian groups A, but not generally for among them, for example, J. H. C. Whitehead, modules. The work of the Japanese mathematician Hassler Whitney, Saunders Mac Lane, and Norman Yoneda showed that an element of Extn(A, B) could Steenrod—who dedicated themselves to building be represented as a long exact sequence of mod- a more adequate armamentarium. Their success in ules (with n intermediate terms): doing this was attested to by the fact that by the end of the 1960s most of those problems had been 0 B E1 E2 En A 0. solved (inordinately many of them by J. F. Adams). → → → →···→ → → Sammy’s contributions appeared for the most All these various examples of the construction part in a series of collaborations. With Mac Lane of new functors as “derived” functors of given he developed the theory of cohomology of groups, ones were at hand for Eilenberg. He saw how they thus providing a proper setting for the remarkable could be used to determine a homological “di- theorem of Hopf on the homology of highly con- mension” for algebraic objects, and he established nected spaces. This led them to the study of the the connection with the Hilbert notion of a syzygy Eilenberg-Mac Lane spaces and thus to a deeper un- in a 1956 paper [3]. This provided the background derstanding of the relations between homotopy and for the influential Cartan-Eilenberg book [1] on homology. Their most fateful invention perhaps homological algebra. This text emphasized how the was that of category theory, responding, no doubt,

derived functors for a module M could be calcu- to the exigencies of algebraic topology but destined

→ → → → lated from any→ “resolution” of M by free mod- to radiate across most of mathematics. ules, a long exact sequence In collaboration with Steenrod, Sammy drained the Pontine Marshes of homology theory, turning 0 M X0 X1 M2 an ugly morass of variously motivated construc- ··· tions into a simple and elegant system of axioms with all Xj free. One simply applies the functor to applied, for the first time, to functors. This was a the resolution with the M term dropped and then radical innovation. Heretofore homology theories takes the homology or cohomology of the result- had been procedures for computing; henceforth ing complex. This effectively generalized the com- they would be mathematical objects in their own putation from specific “bar resolutions” used to de- right. What was especially remarkable was that in fine the cohomology of a group. The ideas of order to achieve this, Sammy and Steenrod un- homological algebra were presented in two pio- dertook to raise the logical level of the things that neering books by Cartan-Eilenberg [1] and Mac Lane might be so regarded. [4]. The Cartan-Eilenberg treatise had a widespread The algebraic structures of the new algebraic and decisive influence in algebra. This again il- topology were proving themselves useful in other lustrates the genius of Eilenberg: If essentially the parts of mathematics: in algebra, representation same idea crops up in different places, follow it out theory, , and even in number the- and find out where it lives. ory. Together with Henri Cartan, Sammy system- atized these structures under the rubric of Ho- mological Algebra, once more raising the level of discourse by introducing such notions as derived Alex Heller functors. I am tempted to insert a parenthesis When I met Samuel Eilenberg in 1947, he was here. This latest innovation brought its authors into introduced as Sammy. He was always referred to conflict with the “establishment” by putting in as Sammy. It would be wrong to speak of him oth- question the very notion of definition, raising a fun- erwise. I was then a student; I promptly became his damental question of the relation between category student. I would like to record what drew me then theory and set theory that has yet to be put de- to Sammy and continued over the years to do so— finitively to rest. Since homological algebra has namely, what I perceived as his radical insistence proved indispensable, the honors lie, I think, with on lucidity, order, and understanding as opposed Cartan and Eilenberg. In any case, the field prolif- to trophy hunting, and his idea of how that un- erated so rapidly that Grothendieck, only a few derstanding was to be achieved. years later, was said to have spoken of their book as “le diplodocus”, regarding it apparently as Alex Heller is professor of mathematics at the Graduate palaeontology. School and University Center, CUNY. His e-mail address The roots of homological algebra lay neverthe- is [email protected]. less in algebraic topology, and Sammy, in collab-

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oration with John Moore, returned to these. They introduced such novelties as differential graded ho- Peter Freyd mological algebra and relative homological algebra Thirty years ago I found myself a neighbor of to provide homes for the new techniques intro- Arthur Upham Pope, the master of ancient Persian duced not only by Sammy and his collaborators but art. He had retired in his nineties to an estate in also by a new generation including Serre, Grothen- the center of the city of Shiraz in southern Iran, dieck, and Adams. Notable among them are the so- where I lived, briefly, across the street. I found an called Eilenberg-Moore spectral sequences, which excuse for what has to be called an audience, and deal with pullbacks of fibrations and with associ- I mentioned that I was a friend of Samuel Eilenberg. ated fiber bundles. “I don’t know him,” he said. “I know of him, of Unfortunately neither Sammy nor his last col- course. How do you know him?” “We work in the same area of mathematics.” laborator, Eldon Dyer, lived to complete their ul- “You’re talking about a different Eilenberg. I timate project of refounding algebraic topology in meant the dealer in Indian art.” the correct—which is to say, homotopical—set- “Actually, it’s the same person. He’s both a ting. Perhaps this project was too ambitious. I mathematician and a collector of Indian art.” learned from Eldon how much agony accompa- “Don’t be silly, young man. The Eilenberg I mean nied even such choices as that of the correct def- is not a collector of Indian art, he’s the dealer in inition of a topological space. Some part of their Indian art. I know him well. He established the book may yet survive, and others are already con- historicity of one of the Persian kings. He certainly tinuing their project piecemeal. is not a mathematician.” As I perceived it, then, Sammy considered that End of audience. 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 In later years even Arthur Upham Pope would have known. In the art world, Eilenberg became uni- definitive clarity that would illuminate its under- versally known as “Professor”. Indeed, if one walked lying order. This was to be accomplished by elu- with him in London or Zürich or even Philadelphia cidating the true structure of the objects of math- and one heard “Professor!”, it was always Eilenberg ematics. Let me hasten to say that this was in no who was being hailed, and it was always the art sense an ontological quest: the true structure was world hailing him. intrinsic to mathematics and was to be discerned If you heard “Sammy!”, you knew it was a math- only by doing more mathematics. Sammy had no ematician. patience for metaphysical argument. He was not a Platonist; equally, he was not a non-Platonist. It might be more to the point to make a different dis- It was complicated, explaining that name. For a tinction: Sammy’s mathematical aesthetic was clas- person who knew him first through his works, it sical rather than romantic. was hard to conceive of him as “Sammy”. And Category theory was one of Sammy’s principal upon meeting him for the first time, it was even tools in his search for mathematical reality. Cate- harder: He was in charge of entire fields of math- ematics—indeed, he had created a number of them. gory theory also developed into a mathematical Whenever he was in a room, he was in charge of subject with its own honorable history and prac- the room, and it did not matter whose room it was. titioners, beginning with Mac Lane and including, Sammy? The name did not fit. notably, F. W. Lawvere, Sammy’s most remarkable But he had to have a name like Sammy. I said it student, who saw it as a foundation for all of math- was hard to explain. Here was one of the most ag- ematics and justified this intuition with such in- gressive people one might ever meet. He would novations as categorical semantics and topos the- challenge almost anything. If a person mentioned ory. Sammy did not, I think, want to be reckoned something about the weather, he would challenge a member of this school. I believe, in fact, that he it: once in California I heard him insist that it was would have rejected the idea that mathematics not weather; it was climate. But somehow it was needed a foundation. Category theory was for him almost always clear: it was all right to challenge him only a tool—in fact, a powerful one—for expand- right back. Aggressive and challenging, but not at ing our understanding. It was his willingness to all pompous. One cannot be pompous with a name search for this understanding at an ever higher level like Sammy. that really set him apart and that made him, in my estimation, the author of a revolution in math- ematics as notable as that initiated by Cantor’s in- Peter Freyd is a professor of mathematics at the Univer- vention of set theory. Like Cantor, Sammy has sity of Pennsylvania. His e-mail address is pjf@ changed the way we think about mathematics. saul.cis.upenn.edu.

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Sammy kept his two worlds, mathematics and who the master faker was and tracking him down art, at something of a distance. But both worlds in his studio, not to berate him, but to congratu- seemed to agree on one thing, the very one that late him. Arthur Upham Pope had insisted upon: Sammy was the dealer. Without question, Sammy loved playing the role After that, Sammy made a point of not build- of dealer. In the days when mathematicians were ing bridges between his two worlds. I recall just one in demand and jobs were easy to come by, Sammy exception. He moved from a conversation about loved to tell about the math market he was going sculpture to one about mathematics. Sculptors, to create. The trade would be in mathematician fu- he said, learn early to create from the inside out: tures: “This one’s done only two lemmas and one what finally is to be seen on the surface is the re- proposition in the last year; the most recent theo- sult of a lot of work in conceptualizing the inte- rem was two years ago; better sell this one at a rior. But there are others for whom the interior is loss.” With his big cigar (expensive) and his big gold the result of a lot of work on getting the surface ring (in fact, a valuable Indian artifact), he could right. “And,” Sammy asked, “isn’t that the case for enter his dealer mode at a moment’s notice. One my mathematics?” always wondered just how many young math- Style is only one part of his mathematics—as, ematicians’ careers were in his hands. of course, he knew—but there are, indeed, won- But his two worlds, mathematics and art, per- derful stories about Sammy, attending only to ceived this role of dealer quite differently. In math- what seemed the most superficial of stylistic ematics we understood that it was a role he loved choices, restructuring entire subjects on the spot. playing, but that he was only playing. His being a Many have witnessed this triumph of style over mathematician was what counted, and he would substance, particularly with students. But the most have been the same mathematician whether or not dramatic example had a stellar cast. D. C. Spencer he played the dealer, indeed, whether or not he gave a colloquium at Columbia in the spring of played —and he did—high-stakes poker. This was 1962, and Sammy decided it was time to demon- not so clear in his other world. strate his get-rid-of-subscripts rule: “If you define it right, you won’t need a subscript.” Spencer, with the greatest of charm—it was for good reason that It was usually frustrating trying to explain to oth- he was already affectionately known as “Uncle ers how Sammy was perceived by his fellow math- Don”—followed Sammy’s orders and proceeded ematicians. Sammy had an unprintable way of say- to restructure his subject while standing there at ing that mathematics required both intelligence and the board. One by one, the subscripts disappeared, aggression. But imagine not knowing how his math- each disappearance preceded by a Sammy-dictated ematics—when he had finished—would totally redefinition. He had virtually no idea of the in- belie that aggression. Imagine not knowing how re- tended meanings of any of the symbols. He was op- markably well-behaved his mathematics always erating entirely on the surface, looking only at the was. Imagine not knowing how his mathematics, shape of the syntax. when he had finished, always seemed preordained The process went on for several minutes, until and how it seemed no more aggressive than, say, Sammy took on the one proposition on the board. the sun rising at its appointed sunrise time. “So now what does that say?” Forty years ago Sammy hoped to turn the study “Sammy, I don’t know. You’re the one making of Indian bronzes into an equally well-behaved all the definitions.” subject. He had already acquired a reputation for So Sammy applied his definitions, and one by being the best detector of fakes in the business, one the subscripts continued to disappear, until and he believed he could axiomatize the process. finally the proposition itself disappeared: it became He even had a provisional list of axioms, and it was the assertion that a thing was equal—behold—to truly an elegant list. itself. A few years later we found ourselves at a small French-style bistro in La Jolla, California. We had been out of touch: there had been an argument “My mother’s father had the town brewery and about mathematical ethics, but somehow we had he had one child, a daughter. He went to the head resolved it; the dinner was something of a cele- of the town yeshiva and asked for the best student,” bration of the resolution. I asked him about his Sammy told me one day. “So my future father be- book on bronzes. came a brewer instead of a rabbi.” “The axioms failed.” Sammy regarded prewar Poland with some af- “What does that mean?” fection. He felt that he had been well nurtured by “It means that I’ve been taken. I bought a fake.” the Polish community of mathematicians, and he He had suspected it only after the work had been told me of his pleasure on being received by Ste- in his bedroom for a few weeks. He had the plea- fan Banach himself, a process of being welcomed sure, at least, of investigating until he found out to the holy of holies, the café in which Banach

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spent his time during the annual Polish math- mathematicians. Sammy’s efforts succeeded for the ematical conferences. By the time Sammy came to language of category theory, and he never aban- the U.S. in his mid-twenties he was a well-known doned his efforts for the theory itself. He was con- topologist. fident that the categorical view would eventually When I questioned him on his attitude about pre- be the standard mathematical view, with or with- war Poland, he answered that one must “watch the out his salesmanship. Its inevitability would be derivative”: Don’t judge just by how good things based not on Sammy’s skills as a dealer but on the are, but by how fast they’re becoming better. theorems whose proofs required category theory. Sammy’s view of Poland since the war was more That was obvious to Sammy. He wanted to make complicated. It was particularly complicated by it obvious to everyone else. what he viewed as its treatment of category the- ory as a fringe subject. Hyman Bass In the late 1950s Sammy began to concentrate his mathematical activities, both research and Sammy visited the University of Chicago for a teaching, on category theory. He and Mac Lane topology meeting while he was department chair had invented the subject, but to them it was always at Columbia. I was then a graduate student, work- an applied subject, not an end in itself. Categories ing with on topics in homologi- were defined in order to define functors, which in cal algebra. So I was already familiar with some of turn were defined in order to define natural trans- Sammy’s work when I first met him and we dis- formations, which were defined finally in order to cussed mathematics. Homological algebra was in- prove theorems that could not be proved before. sinuating itself into and al- In this view, category theory belonged in the main- gebraic geometry through the pioneering work of stream of mathematics. Maurice Auslander and David Buchsbaum (Sammy’s There was another view, the “categories-as- student) and J.-P. Serre. Kaplansky was introduc- fringe” view. It said that categories were defined ing many of my cohorts to this work. in order to state theorems that could not be stated When I graduated in 1959, in a now distant time before, that they were not tools but objects of na- of affluent mathematical opportunity, I contem- ture worthy of study in their own right. Sammy be- plated a year at the Institute for Advanced Study. lieved that this counterview was a direct challenge But Sammy, while I accompanied him to an art to his role as the chief dealer for category theory. dealer in downtown Chicago (an errand whose sig- He had watched many of his inventions become nificance I only later appreciated), persuaded me standard mathematics—singular homology, ob- that it would be better first to launch my profes- struction theory, homological algebra—and he had sional career as a regular faculty member, doing no intention of leaving the future of category the- both research and teaching. That might now seem ory to others. a difficult case to make, but it fit with my own dis- Today the language of category theory has per- position, and, in any case, Sammy had a charismatic meated a good part of mathematics and is treated charm and warm humor that were hard to resist. with some respect. It was not ever so. There were Sammy’s mentoring made me virtually his stu- years before the words “category” and “functor” dent. Columbia’s was a small and intimate de- could be pronounced unapologetically in diverse partment, with such figures as Harish-Chandra, mathematical company. One of my fonder mem- Serge Lang, Paul Smith, Ellis Kolchin, Dick Kadison, ories comes from sitting next to Sammy in the Edgar Lorch, Masatake Kuranishi, , early 1960s when Frank Adams gave one of his first Joan Birman, and, briefly, Heisuke Hironaka, Steve lectures on how every functor on finite-dimen- Smale, Wilfried Schmid, and many others. The de- sional vector spaces gives rise to a natural trans- partment featured some strong personalities, but formation on the K-functor. Frank used that con- Sammy, along with Lipman Bers when he arrived struction to obtain what are now called the Adams somewhat later, set the tone and style of the de- operations, and he used those to count how many partment. Research in topology, algebraic geome- independent vector fields there could be on a try, complex analysis, number theory, and the then sphere. It was not until then that it became per- budding category theory were quite active there. missible to say “functor” without a little snort. Though a faculty member, I functioned much like In those years, Sammy was a one-man employ- a student, learning about both mathematics and ment agency for a fresh generation of math- the intellectual culture of our discipline. ematicians who viewed categories not just as a Over the years my appreciation deepened for the language but as a potentially central mathemati- way Sammy worked and thought about math- cal subject. For the next thirty-five years he went ematics. Though quite accomplished at computa- to just about every category theory conference, and, much more important, he used his masterly ex- Hyman Bass is professor of mathematics at Columbia pository skills to convey categorical ideas to other University. His e-mail address is [email protected].

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tion and geometric reasoning, Sammy was pre- active mind had so much yet to say. Yet he bravely eminently a formalist. He fit squarely into the tra- showed the same good humor and dignity that dition of Hilbert, E. Artin, E. Noether, and Bourbaki; marked his whole life. He leaves us with much to he was a champion of the axiomatic unification that treasure, even while we miss him. so dominated the early postwar mathematics. His philosophy was that the aims of mathematics are References to find and articulate with clarity and economy the [1] H. CARTAN and S. EILENBERG, Homological algebra, underlying principles that govern mathematical Princeton Univ. Press, Princeton, NJ, 1956. phenomena. Complexity and opaqueness were, for [2] S. EILENBERG, Topological methods in abstract algebra: him, signs of insufficient understanding. He sought Cohomology theory of groups, Bull. Amer. Math. Soc. 55 (1949), 3–37. not just theorems, but ways to make the truth [3] ———, Homological dimension and syzygies, Ann. transparent, natural, inevitable for the “right think- Math. (2) 64 (1956), 328–336. ing” person. It was this “right thinking”, not just [4] S. MAC LANE, Homology, Springer-Verlag, Berlin, 1963. facts, that Sammy tried to teach and that, in many domains, he succeeded in Some Ph.D. Students of Samuel Eilenberg teaching to a Kuo-Tsai Chen (1950) whole genera- Alex Heller (1950) tion of math- ematicians. David Buchsbaum (1954) In some (1954) ways Sammy Kalathoor Varadarajan (1954) seemed to F. (1963) have a sense of the structure Harry Applegate (1965) of mathemati- Estelle Goldberg (1965) cal thinking Myles Tierney (1965) that almost George A. Hutchinson (1967) transcended specific sub- Jonathan M. Beck (1967) ject matter. I Stephen C. Johnson (1968) remember the Albert Feuer (1974) uncanny sen- Chang-San Wu (1974) sation of this on more than Martin Golumbic (1975) one occasion Alan Littleford (1979) when sitting next to him in department colloquia. The speaker was exposing a topic with which I knew that Sammy was not particularly fa- miliar. Yet a half to two thirds of the way through the lecture, Sammy would accurately begin to tell me the kinds of things the speaker was going to say next. Though his mathematical ideas may seem to have a kind of crystalline austerity, Sammy was a warm, robust, and very animated human being. For him mathematics was a social activity, whence his many collaborations. He liked to do mathematics on his feet, often prancing while he explained his thoughts. When something connected, one could read it in his impish smile and the sparkle in his eyes. He was engaged with the world in many ways, a sophisticated and wise man who took a refined pleasure in life. His was a most satisfying and in- spiring influence on my own professional life. After his stroke, it was painful to see Sammy, frail and gaunt and deprived of speech when his still

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