Mac Lane, Saunders

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MAC LANE, SAUNDERS

(b. Taftville, Connecticut, 14 August 1909; d. San Francisco, California, 13 April 2005),

mathematics, category theory, mathematics education.

Mac Lane in his long life made powerful and lasting contributions to world mathematics, notably by crystallizing with Samuel Eilenberg the concepts of category, functor, and natural transformation, and then extensively developing and applying them. Those concepts have become indispensable to twentieth- and twenty-first-century thinking about geometry, algebra, and logic and have a growing simplifying influence on analysis, statistics, and physics. One of the few universal mathematicians, Mac Lane was a towering figure because of his enormous work in research and teaching, but his career was also marked by a persistent struggle to bring about change and the acceptance of new ideas through participating in organizations whose tendency was rather to uphold the status quo. Mac Lane’s doctoral students, including Irving Kaplansky, John Thompson, Michael Morley, and Robert Solovay, played important roles in twentieth century mathematical research. He was very active in the Mathematical Association of America, the American Mathematical Society, the National Academy of Sciences (NAS), and in the International Mathematical Union. He received numerous prestigious prizes for scientific achievement, including the National Medal of Science in 1989; in 1972 he was named an Honorary Fellow of the Royal Society of Edinburgh.

Early Life and Career . Mac Lane was 15 years old when his father died and he went to live with his grandfather. His father and grandfather were both pastors in the Congregationalist Church, and Mac Lane admired both of them for their courage in preaching nonconformist views such as Darwinism and pacifism, but he could never accept their religious ideas. An uncle financed his study at Yale University, where he graduated in 1930 with the highest academic standing in the history of the university, but he was not elected to the notorious Skull and Bones. He earned his master’s degree in 1931 at the University of Chicago, where Eliakim H. Moore counseled him to go to Germany; he became the last American to earn a mathematics doctorate at the University of Göttingen of David Hilbert, Emmy Noether, Hermann Weyl, and Paul Bernays. He hid his copy of Das Kapital to prevent its being burned when he witnessed the Nazis taking control in January 1933. After his return to the United States, he taught mathematics at Cornell, the University of Chicago, and for ten years at Harvard University. In 1947 he became a professor at the University of Chicago, where his vigorous and inspired teaching continued well after his retirement in 1982.

Activities in Professional Organizations . From 1943 to 1981 Mac Lane was very active in various professional organizations. He served as director of Applied Mathematics at Columbia University from 1943–1945, on leave from Harvard, as part of the war effort. He was president of the Mathematical Association of America in 1951–1952, and began efforts at the national level to reform the teaching of mathematics. On 1 February 1952 he issued the directive to all sections of the MAA that minorities must have equal access to the academic and social functions of the association, contrary to the previous practice of some of the sections. From 1952–1958 he was chairman of the mathematics department at the University of Chicago, succeeding his friend Marshall Stone. In 1973–1974 he was president of the American Mathematical Society and was vice president of the National Academy of Sciences until 1981. During the eight years at the NAS he devoted much of his energy to chairing the Reports Review Committee, and was from time to time compelled to issue forceful calls for greater scientific seriousness—for example, in connection with a report on the effects of the military use of poisons in Vietnam. He did the same in other contexts, which, as one would expect, led to a mixed popularity.

Mac Lane strove valiantly to promote that closer unity between teaching and research that was so much the essence of his own mathematical life. To advance that purpose, he urged a merger of the professional societies, but succeeded only in creating one umbrella committee, the Joint Policy Board for Mathematics.

Through his organizational initiatives at the national level in the mid-1950s, he had applied his international mathematics experience to courses for high school teachers, which he and other active mathematicians taught. In the early 1960s, however, hopes for a progressive new math were frustrated when university presidents and government agencies cut the funding for these courses. Channelling energies into a retrogressive “new math,” various authorities made organizational decisions that, in Mac Lane’s view, tended to steer high school teachers towards outmoded pedagogical theories, instead of scientific thinking and mathematical content. Mac Lane’s efforts to promote improved conditions for scientific research and education achieved only modest results, in spite of the great amount of time he spent in Washington. That experience contributed to his later analysis of what he saw as grave flaws in the methods for arriving at science policy in the federal government and the American university system.

Influential Textbooks . Fortunately, Mac Lane’s energies were not entirely devoted to organizational efforts, but also to his own fruitful research and teaching and especially to the relation between them. His textbooks A Survey of Modern Algebra (1941, with Garrett Birkhoff), Homology (1963), Categories for the Working Mathematician (1971), and Sheaves in Geometry and Logic (1991, with Ieke Moerdijk) are still widely used in the early twenty-first century. Mac Lane’s book with Birkhoff made Bartel van der Waerden’s Moderne Algebra (1930) accessible to English-speaking undergraduates. The Survey was fundamental to the education of several generations, and Mac Lane rewrote the 1967 edition in order to respond explicitly to the growing need for the learning of category theory. All four of these textbooks fundamentally contributed to bringing new abstract research to students at the time when they needed to learn it.

Homological Functors and Abelian Categories . Forceful personality and energetic perseverance were not the only attributes that made Mac Lane so prominent; rather, the primary reason is that his central ideas were, and have remained, correct. He accurately summed up the achievements of the previous generation and passed them on, forever transformed in clarity and applicability. This process can be clearly discerned in the cases of homological functors and Abelian categories.

By 1940 Mac Lane and his friend Sammy Eilenberg had each made significant contributions to their respective fields of algebra and topology, and thus, through their collaboration on the challenging problems of Heinz Hopf and Norman Steenrod in algebraic topology, could gain access to the rich social patrimony of several centuries of mathematical development. Reflected through that access was sufficient knowledge of the forms of results, and especially of some main modes of the development of ideas, so that they were able to concentrate and isolate the explicit concepts that they called category, functor, and natural transformation. These explicit concepts were so correct as a reflection of the essence of various aspects of mathematical content and motion that they immediately provided a source of structures whose properties could be studied with fruitful results for mathematics in general. The concept of functor, almost immediately after its discovery by Eilenberg and Mac Lane in 1942 (and expounded by them in 1945) provided a structure to which axioms and deduced theorems could be applied; specifically, the axioms announced by Eilenberg and Steenrod in 1945 (and expounded by them in 1952) clarified the previous proliferation of geometrical constructions known as homology theories, and in turn made possible still richer such theories. Functorial homology theory became a cornerstone of the still ongoing research in algebraic topology. The axiomatic method could similarly be applied to categories themselves, as was then exploited by Mac Lane (in 1948, expounded in 1950). He captured the essence of linear algebra via the axiom that products and coproducts (which themselves can only sensibly be defined by categorical means) coincide in certain categories. Often such categories enjoy the internal representability of the solutions of any equation they contain; if they satisfy certain “exactness” conditions, such categories are called “Abelian” after the great Norwegian mathematician Niels Abel (1802–1829).

These Abelian categories quickly served their purpose as another cornerstone of algebraic topology and were host to a new branch of linear algebra that became known as homological algebra. Homological algebra had undergone extensive development in the collaboration of Eilenberg and Mac Lane in the late 1940s and early 1950s on the homology of groups. Over the next decades, Abelian categories underwent deep development by David Buchsbaum, Alexander Grothendieck, and Jean-Pierre Serre, and further by Maurice Auslander, Michael Barr, Peter Freyd, Peter Gabriel, Alex Heller, Barry Mitchell, Stephen Schanuel, Jean-Louis Verdier, Nobuo Yoneda, and others. Philosophically, those developments meant in particular that methods previously conceived as applying only to constant quantities could be extended to apply also to variable quantities, with powerful results.

Adjoint Functors . Axiomatic algebraic topology and homological algebra can both be described as having been new category theory, arising in a sense entirely within category theory, but in response to the needs of application. The most important instance of this phenomenon, it is generally agreed, is Daniel Kan’s 1958 discovery of adjoint functors (which, in retrospect, were implicit in the Eilenberg-Mac Lane 1945 paper). This concept united a wealth of old and new examples, again exploiting the susceptibility of the appropriate structures to restricting properties that, as axioms, have powerful consequences and serve as a guide to further constructions, conjectures, and theorems. The particular problems occupying Kan concern the relation between the qualities of combinatorial homotopy theory and the qualities of quantities arising in differential vector calculus (a relation that lies at the basis of the finite element method in applied electro-magnetism, for example). Kan discovered that a functor from one category to another might be so special as to have another uniquely determined functor in the opposite direction that, while not actually inverting it, is the “best” approximation to an inverse (in either a left- or a right-handed sense). Typically, one of the two functors is so obvious that one might not have mentioned it, whereas its resulting adjoint functor is a construction bristling with content that moves mathematics forward.

One of many examples, whereby the use of adjoint functors helps old constructions become much more explicit and clear, is a construction which had played a key role in Mac Lane’s research in Galois theory and explains the realization of conjugation on the complex numbers (the process of negating the generator i) as an inner automorphism (by j) of the larger enveloping algebra of Hamilton’s quaternions. There is a similar realization of the mechanical flow on a phase space via an inner derivation by a Hamiltonian element in an algebra of operators. The conjugation example involves a two-element group in two different roles, and the mechanical example similarly involves an infinitesimal group of time translations. The “inner realization” construction is the left adjoint of the functor, determined by a given group G, which to every algebra A with a given representation of G by multiplication in A, assigns the action of G on A defined by

ag = g-1ag.

The left adjoint to this process applies to any algebra with a given action of G on it, enlarging it in an optimal way to make the action inner; the resulting algebra, which contains this inner action of G in it, is usually non-commutative, even if the given one was commutative.

Adjointness and Logic . Naturally, Kan’s discovery spurred a succession of new leaps forward within category theory in response to its relation with applications. Some of these were intimately related with Mac Lane’s longstanding interest in logic and set theory. Already during the 1931 interval between his Yale degree and his studies at Göttingen, he had studied at the University of Chicago with Eliakim H. Moore. Moore too was a very strong personality who, coming from algebra, had, like Mac Lane, a burning desire and specific proposals to unify mathematical research and to reform mathematical teaching to that end. Mac Lane had taken up from Moore the quest to axiomatize set theory. Mac Lane’s thesis at Göttingen had resulted from intense discussions with the set theorist Paul Bernays concerning the possibility of a formalized logic that could actually be used to guide mathematical proofs. During the next 30 years, however, Mac Lane did not concentrate his research on set theory or logic, although he did make valiant organizational efforts, promoting the formation of logic clubs among undergraduates and reviewing for the Journal of Symbolic Logic. He was pleased in the early 1960s when it became apparent that logic and set theory, insofar as they are mathematically relevant, can be characterized axiomatically as specific interlocking systems of adjoint functors: specifically

propositional logic symbolically presents parts of a universe of discourse in terms of pairs of operations like G & () and G implies (), related by rules of modus ponens and deduction which say no more than that those operations are adjoint;
predicate logic treats, moreover, parts of several universes related by maps (for example projection maps), where the fundamental categorical process of composition is exemplified by substitution along the map (representing inverse image) that has a left adjoint, namely existential quantification along the map (representing direct image); and
higher-order logic treats, moreover, a system of several universes wherein, for any universe G, there are adjoint functors creating as new universes the G-cylinder and G-figure universes (also known as function types; part of the adjunction property had been called lambda-conversion by Alonzo Church).

Set theory itself was quickly seen in a new light via adjointness; after all, the adjointness of function types expressed a fundamental transformation that had been used in functional analysis (and in its embryo, calculus of variations) for 250 years, and the belief that Georg Cantor’s set theory would have an important role in analysis (as expressed at the first International Congress of Mathematicians in 1897 by Jacques Hadamard) had sprung from the intense work at that time which was bringing functional analysis to the light of day. Quickly overcoming his initial skepticism, Mac Lane recognized the decisive importance, for set theory and logic and their relationship to mathematics, of the explicitly adjoint character of these operations. He sprang into action: He made sure that the basics were published by making F. William Lawvere’s Elementary Theory of the Category of Sets available through the University of Chicago library, and ensuring that an announcement of that work appeared in the Proceedings of the National Academy of Sciences (1964); he wrote expositions himself for all kinds of audiences; and he engaged in published polemics with recalcitrant set-theorists, right up to the new millennium (2000).

The Geometrical Use of Category Theory Gives Rise to New Category Theory . During the same period, Alexander Grothendieck was creating the new foundation for algebraic geometry, which was also based on categories and adjoint functors. He realized that the variable linear algebra that he (following Mac Lane) had developed in the late 1950s, is best viewed as an additional structure on a nonlinear kind of category, such as set-valued sheaves or analytic spaces. This led to the crystallization of a new kind of categories, which he called toposes because they were the brave new manifestation of the science of situation. (The Greek term was apparently chosen to signify a qualitative deepening of the analysis situs of Henri Poincaré.) These situations serve as the domains of variation for variable sets. In the 1970s students in Paris, at Harvard, and at other centers had to struggle to learn the topos theory through the 2000 pages (written with the help of Michael Artin, Jean-Louis Verdier, and others) that only the brain of a Grothendieck could really encompass.

Grothendieck retired suddenly in 1970, but the new algebraic geometry continued to develop. Meanwhile a simplified form of the topos theory had already sprung up, with motivations from continuum mechanics, but with new applications to set theory and logic. (That work, achieved in collaboration with Myles Tierney, was presented in 1970 by Lawvere at the International Congress of Mathematicians in Nice, France.) For a time those two trends, algebraic geometry and the new topos theory, were very slow in learning from one another, in contrast with the situation in the 1950s when category theory had been revolutionizing not only the framework of mathematics, but also its practice. There were logicians who, like Galileo’s colleague, refused to look into “the telescope” that was provided by books on the topos-theoretic simplification of logic; there were also algebraic geometers who dismissed the modern topos geometry as “mere logic.” Mac Lane did not despair in the face of these difficulties of communication and lack of mutual understanding.

That was precisely the sort of wrong that Mac Lane knew how to begin to set right: girding himself anew in the middle of his seventh decade, he set off for Cambridge where his lectures inspired Peter Johnstone to write the first book on the new topos theory (1977). In the middle of Mac Lane’s ninth decade, yet another book appeared, the result of his collaboration with Ieke Moerdijk of the University of Utrecht in the Netherlands; this latter book, Sheaves in Geometry and Logic (1992), was necessary because the books that had appeared in the intervening twenty years on this emerging subject had still not covered all the varied developments. The book with Moerdijk shows clear traces of the hand of the master expositor: the systematic use of presheaf toposes and Kan adjoints and the careful exposition of the relation between the combinatorial topology of 1950 and the internal role of the adjointness formulation of logic.

Continuing Influence . Algebraic geometry, complex analysis, universal algebra, logic, and in fact any given field in mathematics has as neighbors other growing fields wherein the categorical method is already indispensable. Homotopy theory, for example, uses the Eilenberg-Mac Lane spaces (introduced in the mid-1940s) and is often based on Daniel Quillen’s axioms, built on the categorical work of Peter Gabriel and Michel Zisman. As another example, the closed and enriched categories of Samuel Eilenberg and Gregory M. Kelly have led to a rejuvenated study of metric spaces and of generalized logic, unexpected when they were introduced in 1965 as the definitive solution to the problem of signs in algebraic topology. The force of the streams of these neighboring developments will eat away any resistance of remaining “islands” to the unification that the progress of mathematics requires.

When his formulation of the homology of rings was taken up and developed in 1961 by Umeshachandra Shukla, Mac Lane remarked that “as always, it is a pleasure to see how new ideas spread” (2005, p. 222). When visiting Soviet Georgia in 1987, he found to his delight a group of a dozen devoted and enthusiastic researchers in category theory and concluded, “it is remarkable to see how abstract mathematical ideas have international resonance” (2005, p. 331).

The enormous power of a correct idea cannot always be foreseen. Mac Lane found it hard to believe that already by the late 1950s, the fundamentals of category theory had penetrated into the midwestern farmlands, enabling fledgling students there to discuss and dream about its intimations of powerful unifying developments. The correctness of these explicit ideas was an electrifying inspiration to us then, and remains an enduring inspiration to scientific progress.

BIBLIOGRAPHY

WORKS BY MAC LANE

With Garrett Birkhoff. A Survey of Modern Algebra. New York: Macmillan, 1941.

With Samuel Eilenberg. “General Theory of Natural Equivalences.” Transactions of the American Mathematical Society 58 (1945): 231–294.

“Groups, Categories, and Duality.” Proceedings of the National Academy of Sciences, U.S.A. 34 (1948): 263–267.

Homology. Berlin: Springer-Verlag, 1963.

With Garrett Birkhoff. Algebra. New York: Macmillan, 1967.

Categories for the Working Mathematician. New York: Springer-Verlag, 1971.

“Origins of the Cohomology of Groups.” Enseignement Mathématique 24 (1978): 1–29.

Selected Papers. Edited by I. Kaplansky. New York: Springer-Verlag, 1979.

With Samuel Eilenberg. Collected Works. Orlando, FL: Academic Press, 1986a.

Mathematics: Form and Function. New York: Springer-Verlag, 1986b.

“Concepts and Categories in Perspective.” In A Century of Mathematics in America, Part 1, edited by Peter Duren. Providence, RI: American Mathematical Society, 1988.

With Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. New York: Springer-Verlag, 1992.

“Contrary Statements about Mathematics.” Bulletin of the London Mathematical Society 32 (2000): 527.

Saunders Mac Lane: A Mathematical Autobiography. Wellesley, MA: A. K. Peters, 2005.