The Times 3 May 2005 Features Saunders Mac Lane;Obituary;The Register 1,109 words 50 English Saunders Mac Lane, mathematician, was born on August 4, 1909. He died on April 14, 2005, aged 95. Mathematician who helped to create the language of category theory and then made it indispensable. IT IS no exaggeration to say that Saunders Mac Lane was one of the most influential mathematicians of the 20th century. The former professor at the University of Chicago, he was widely known as the creator, with Samuel Eilenberg, of the language of category theory, now an indispensable tool in many fields of mathematics, and still finding new applications in other fields, notably computer science and theoretical physics. Saunders Mac Lane was born in 1909 in Norwich, Connecticut. His father, Donald Mac Lane, was a Congregational minister; his mother, Winifred nee Saunders was a school teacher. His father died when Saunders Mac Lane was only 15, and after he graduated from high school in Leominster, Massachusetts, it was an uncle who in 1926 sent him to Yale. He graduated from Yale in 1930, and then moved to the University of Chicago where he obtained a masters degree in 1931. In that year he met Dorothy Jones, who was reading economics at the same university, and they married in 1933. Mac Lane then went to Gottingen to work on his PhD thesis. In those days, it was not uncommon for talented mathematicians from the US to study in Europe, and Gottingen was a world centre of mathematics. There Mac Lane was able to attend lectures by the mathematical elite of the time, including David Hilbert, Gustav Herglotz, Hermann Weyl and Emmy Noether. One of the developments that was taking place in Gottingen was the shaping of a "modern algebra", of which Emmy Noether was one of the main proponents. The political climate did not allow Mac Lane to stay long in Germany, however, and he had to hurry to complete his thesis in mathematical logic, under the supervision of Hermann Weyl and Paul Bernays. He returned to the US in the summer of 1933 to take a one-year job at Yale. After stints as an instructor at Harvard, Cornell and Chicago, he went to Harvard as an assistant professor in 1938; he was appointed to full professor in 1946. In 1947 he accepted an offer from the University of Chicago, where he was to spend the rest of his long career. He served as chairman of the mathematics department there from 1952 to 1958, and was appointed Max Mason Distinguished Service Professor in 1963, and professor emeritus in 1982. On his return from Germany to the US, Mac Lane's research interests had shifted from logic to algebra. Together with Garrett Birkhoff, he took it upon himself to spread the modern algebra of Emmy Noether on the American continent, and in 1941 Birkhoff and Mac Lane published their celebrated Survey of Modern Algebra, the first English-language text on the subject. The book has seen many editions and has influenced the lives of many generations of mathematics students in North America and Europe alike. In his own research, Mac Lane had moved on and, with Samuel Eilenberg and others, he developed an entirely new field of algebra, designed to provide the methods for a quantative, "algebraic" study of geometric objects, called homological algebra. They discovered a class of mathematical objects -now known as Eilenberg-Mac Lane spaces -which combine algebraic and geometric features in an elegantly miraculous way and which continue to play a central role in the field. In his work on homological algebra, Mac Lane showed the rare combination of being capable of doing long and complicated calculations (using new concepts to be developed simultaneously), and of being a clear teacher and expositor at the same time. It was in the course of this work with Eilenberg that the two mathematicians became increasingly aware of the lack of tools, even of a language, that would enable one to move from a domain such as geometry, to another such as algebra, in a way that not only relates geometric objects to algebraic ones, but also translates -in an informative way -relations between different geometric objects into algebraic relations. This led to their development of category theory. The first publication on category theory by Mac Lane and Eilenberg in 1945 was received by the mathematical community with mixed feelings, as it seemed merely to express known facts in a new language. But gradually the force and wide applicability of the new language became clear, and now many important modern insights in mathematics cannot even be formulated without this language. Moreover, it turned out to be an effective vehicle, not only for exhibiting existing as well as newly discovered relations between different areas of mathematical research, but also for dealing with relations between mathematics and other disciplines. Mac Lane wrote two influential textbooks on these subjects which he partly created himself: Homological Algebra, first published in 1963, and Categories for the Working Mathematician, first published in 1971 and still the prevailing textbook in the field. Mac Lane was responsible for significant breakthroughs in mathematical research and wrote several books. Moreover, he was an inspiring teacher to many students, and served the mathematical community as a president of the American Mathematical Society, as a member of the National Science Board and in various other significant functions. He was awarded the National Medal of Science in 1989. In addition, he was a gifted teacher, who was able at the same time to teach, to challenge and to listen to young students. He was an enthusiastic hiker, leading the troops at mathematical conferences when he was already well into his seventies. He had many interests besides mathematics, notably English poetry and philosophy, and it was a great pleasure for colleagues, especially younger ones, to share his seemingly endless discussions about how (and how not) to present mathematical concepts. Mac Lane would argue fiercely for his own point of view, but would always concentrate on the content of the matter, and never impose his authority or seniority. And when the conversation went beyond mathematics Mac Lane would discuss philosophy, from Hegel and Heidegger to Kant and beyond, explaining at length why, for examples, Wittgenstein's view on the philosophy of mathematics was all wrong, or telling colourful stories about how mathematics was before the war, about Hermann Weyl in Gottingen, or about his encounters with Bertrand Russell at a seminar at Harvard. After the death of his wife Dorothy in 1985, Mac Lane married Osa Mac Lane-Skotting. She and his two daughters by his first marriage survive him. [PJF: The 1945 paper is always cited as the first category paper, but it should be noted that the first appearance of functors and natural transformations was 3 years earlier:"Natural isomorphisms in group theory" Proc. Nat. Acad. Sci. U.S.A. 28, (1942). 537--543. [Andre Weil's review:] A vague idea of covariance and contravariance is often met with in group- theory, topology, etc.; that is, one feels that the character-group is contravariant to the group, that the homology and co-homology groups of a complex are, respectively, covariant and contravariant to the complex. This is of special importance in the building up of limits of direct and inverse systems ("projective" and "inductive" limits) of groups, spaces, etc. The authors have succeeded in finding for this a precise definition, which is likely to be helpful in classifying and systematizing known results and also in looking for new relations between groups. In this note, they give a brief sketch of their method, for groups only. The main idea is that of a functor, which will best be explained by an example: for them, the definition of the character-group to an Abelian group G is only one half of the definition of a functor, which they call Ch(G), the other half being the (obvious) rule by which any homomorphism of G into another group H determines a homomorphism of the character-group of H into the character-group of G. Generally speaking, a functor, associated with some groups G_1, G_2,... consists of the definition of some associated group, together with a rule indicating that the latter behaves in a certain prescribed fashion under homomorphic transformations affecting G_1,G_2,.... Examples are given to illustrate this concept; in particular, the authors use it to derive some interesting relations concerning Whitney's "tensor-product" of groups, and clarify the nature of the latter.