Saunders Mac Lane

Saunders Mac Lane was the joint creator - with Samuel Eilenberg - of the subject of category theory, which permanently changed the way in which mathematicians view the foundations of their subject.

Saunders Mac Lane, mathematician: born Taftville, Connecticut 4 August 1909; staff, Mathematics Department, University of Chicago 1947-63, Max Mason Distinguished Service Professor 1963-82 (Emeritus); married 1933 Dorothy Jones (died 1985; two daughters), 1986 Osa Segal (née Skotting); died San Francisco 14 April 2005.

Saunders Mac Lane was the joint creator - with Samuel Eilenberg - of the subject of category theory, which permanently changed the way in which mathematicians view the foundations of their subject.

But he will also be remembered by many generations of mathematics students as the co-author, with Garrett Birkhoff, of A Survey of Modern Algebra (1941), which for at least 30 years remained the definitive English-language treatment of abstract algebra. And the more select company of those who had the good fortune to encounter him as a teacher will remember one of the most forceful and inspiring lecturers imaginable.

Mac Lane was born in Taftville, Connecticut, in 1909. His father and grandfather were both Congregationalist ministers, and his nonconformist ancestry could be detected in his strong sense of duty and service (allied to a sense of humour, not to mention a dress sense, that was decidedly un-Puritan). His father died when he was 15, and an uncle made it possible for Mac Lane to go to Yale, where he excelled in mathematics. After a year of postgraduate work in Chicago, he went to Göttingen to study for a PhD in mathematical logic under the supervision of Paul Bernays. On returning to the United States, he spent the next five years in fixed-term appointments at Yale, Harvard, Cornell and Chicago, before becoming an assistant professor at Harvard in 1938.

Although his PhD thesis had been in logic, Mac Lane's early published papers were in algebra (particularly valuation theory), a subject which he had imbibed deeply at Göttingen from Emmy Noether, the "mother of modern algebra", and her students. It was at Harvard that he met Birkhoff, who like him had travelled to Europe (in his case Cambridge) to do research, and they agreed to collaborate on writing an English-language account of the new developments in algebra which they had both encountered there. At that time the only available textbook on "modern algebra" was Bartel van der Waerden's Moderne Algebra (1931), which was fairly tough going, even for those who could read German (an English translation eventually appeared in 1949). Birkhoff and Mac Lane, in contrast, produced a classic which remained a standard reference for undergraduate algebra courses for more than three decades, and is still in print today.

It was in 1941, the year the Survey was published, that the Eilenberg-Mac Lane collaboration began. Eilenberg, a Polish Jew trained in the formidable Polish school of topologists, had arrived in the United States two years earlier and was then at the University of Michigan. Mac Lane visited Michigan to lecture on his current research in group extensions; Eilenberg recognised the similarity of Mac Lane's calculations to ones he was encountering in algebraic topology, and suggested to Mac Lane that they collaborate. The partnership lasted 14 years, and produced 15 major papers which changed the direction of 20th-century mathematics.

The common calculations gave rise to the subject of homological algebra, a subject which was created independently (thanks to wartime communication difficulties) by researchers in Europe. But Mac Lane and Eilenberg went further. In seeking to provide a sound conceptual framework for the subject, they invented the notions of category and functor. These notions were slow to gain acceptance (the first Eilenberg-Mac Lane paper on categories was nearly rejected by the Transactions of the American Mathematical Society) on account of their seeming lack of content: for a decade or so, category theory was derided by other mathematicians as "abstract nonsense". But in time the substantial new advances made possible by the categorical way of thinking about mathematics won it acceptance: it has by now become an indispensable part of the vocabulary of the great majority of pure mathematicians (and, increasingly, of researchers in theoretical physics and computer science).

Apart from a period at Columbia University in 1944-45, when he worked on problems in aerial gunnery as part of the US war effort, Mac Lane remained at Harvard until 1947, when he was recruited by Marshall Stone to join the faculty of the University of Chicago. He spent the rest of his career at Chicago, becoming Max Mason Distinguished Service Professor in 1963 and Professor Emeritus in 1982. His distinction was recognised by honorary degrees from several universities, and by the award of the National Medal of Science in 1989.

Mac Lane's sense of duty, and his real concern for the health of the academic profession, led him to take on onerous administrative responsibilities. But he found administration uncongenial in comparison with actually doing mathematics. He was a born teacher: both as a forceful and challenging lecturer, and also as an adviser of research students. His list of PhD students is both long and distinguished: significantly, it includes a good many mathematicians who went on to eminence in fields quite different from his own. His last student received his degree when Mac Lane was 87.

He greatly enjoyed the company of younger colleagues, and remained an active participant in international conferences right up to his 90th year. His physical energy astonished his younger colleagues: on the "conference hike", he would always be among the leaders, dressed in tartan trousers (the MacLean tartan, of course) and wielding a heavy stick.

In addition to the best-selling A Survey of Modern Algebra, Mac Lane was author or co-author of five books. His accounts of homological algebra (Homology, 1963) and category theory (Categories for the Working Mathematician, 1971) both carry the authority of having been written by a founder of the subject; a second algebra textbook with Birkhoff (Algebra, 1967) was a pioneering attempt to let the new ideas of category theory "trickle down" into the undergraduate curriculum; a book on philosophy of mathematics (Mathematics: form and function, 1985) was unconventional but widely admired; and his last book (Sheaves in Geometry and Logic, 1992), written jointly with Ieke Moerdijk, was a textbook on topos theory, one of the major new developments in mathematics that category theory made possible.

Sadly, he did not live to see the publication of A Mathematical Autobiography, which is due to appear in a few weeks' time.

Peter Johnstone