An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

The University of Chicago’s Saunders Mac Lane, one of the most influential American mathematicians of the 20th century and a recipient of the National Medal of Science, died Thursday, April 14, in San Francisco after a long illness. He was 95.

“He was one of the most important figures in the University of Chicago Mathematics Department, or indeed in American mathematics,” David Eisenbud wrote of Mac Lane in the preface of the latter’s autobiography, which will be published in late May by A K Peters Ltd. Eisenbud, who received his Ph.D. in Mathematics from the University of Chicago under Mac Lane’s supervision in 1970, is the president of the American Mathematical Society and director of the Mathematical Sciences Research Institute in Berkeley, Calif.

Peter May, Professor in Mathematics at the University of Chicago, described the multifaceted Mac Lane as “extremely energetic, dynamic, clear-headed, opinionated, a raconteur.” In his research, “he was extraordinarily perceptive and original, and he was especially strong as a philosopher of mathematics,” May said.

“With Sammy Eilenberg he created a new way of thinking about mathematics. In a landmark 1945 paper, they introduced and named the concepts of ‘categories,’ ‘functors’ and ‘natural transformations.’ The language they introduced there transformed modern mathematics,” he said. “In fact, a very great deal of mathematics since then would quite literally have been unthinkable without that language.”

F. William Lawvere, Emeritus professor of mathematics at the State University of New York at Buffalo, said, “Category theory is still exploding in its influence after 60 years, illuminating and guiding the development of practically every one of the many varied fields of mathematics.”

Category theory was first developed as a language to describe transformations from one area of mathematics into another, although it later developed into a field of study in its own right. Said May, “Mac Lane was one of the pioneers of algebraic topology, a subject in which one transforms, or describes, spatial structures, which one first sees in terms of shapes—spaces with holes, like a doughnut, say—into algebraic structures, with addition and multiplication, in which one can do calculations.”

The development of category theory and algebraic topology was accompanied by the development of another subject, homological algebra, a kind of algebra that plays a prominent role in algebraic topology and other branches of mathematics. “Mac Lane played a major role in establishing the foundations of these and related areas of modern mathematics, and he wrote famous texts that are some of the most readable accounts of these fields,” May said.

Category theory was at first perceived by some mathematicians as too abstract for practical mathematics, May said. Consequently, Mac Lane titled his introductory book to the field Categories for the Working Mathematician.

Peter Johnstone, professor of the foundations of mathematics at the University of Cambridge, also lauded Mac Lane for his legacy as a mathematical historian. “In the many papers he wrote on historical topics in his later years, he has left a unique body of material for future historians of 20th-century mathematics, written by someone who was there at the time and who knew what it was like to be working at the cutting edge of mathematical research,” Johnstone said.

Mac Lane also steered national science and mathematics policy through his work on numerous boards, and guided dozens of students to mathematical careers.

In the support of scientific research, Mac Lane served as vice president of the National Academy of Sciences and the American Philosophical Society, and as president of the Mathematical Association of America and the American Mathematical Society. As president of the Mathematical Association of America in the 1950s, he began the first activity of that organization toward improvement in the teaching of modern mathematics.

He was a member of the National Science Board from 1974 to 1980, providing science policy advice to the U.S. government. In 1976, he led a delegation of mathematicians to the People’s Republic of China to examine the conditions affecting the development of mathematics in that nation.

“Saunders’ principal characteristic was his immensely strong sense of duty and service,” Johnstone said. “He felt it was his duty to the profession to which he was privileged to belong, even though I’m sure he found it less congenial work than actually doing mathematics.”

For a time there were three members of the University of Chicago Mathematics Department who had received their Ph.D.s under Mac Lane’s supervision. The first of the three was Alfred Putnam, who studied under Mac Lane at Harvard University in 1942. The other two were John Thompson, who received his Ph.D. in 1959, and Arunas Liulevicius, who did likewise in 1960, both from the University of Chicago. Thompson received the Fields Medal, sometimes referred to as the Nobel Prize of mathematics, in 1970.

Another prominent student of Mac Lane’s, Irving Kaplansky, also served on the University of Chicago faculty. Kaplansky was Mac Lane’s first Ph.D. student, receiving his degree at Harvard in 1941. Kaplansky served as Chairman of Chicago’s Department of Mathematics and as director of Berkeley’s Mathematical Sciences Research Institute, and is a member of the National Academy of Sciences.

Mac Lane supervised 39 Ph.D. students at Chicago, a record surpassed by only three other professors in the history of the University’s Mathematics Department. His last Ph.D. student, Steve Awodey, completed his degree in logic in 1997. Then in his 80s, Mac Lane was still energetic, always insisting upon taking the stairs to his third-floor office in the Ryerson Physical Laboratory, Awodey recalled.

Mac Lane was “a link to a past world of mathematics that’s gone now,” said Awodey, an associate professor in philosophy at Carnegie Mellon University. Mac Lane studied logic and mathematics as a Ph.D. student under several world-renowned mathematicians in the early 1930s in Göttingen, Germany, including David Hilbert, Emmy Noether and Hermann Weyl. At the time Göttingen was the world’s center of mathematical research, and Mac Lane often told stories of the great figures he encountered there.

Mac Lane was born in Norwich, Conn., on Aug. 4, 1909. He earned his bachelor’s degree from Yale College in 1930, and his master’s from the University of Chicago in 1931. His first scientific paper was published during this period, in physics. It was co-authored by Irving Langmuir, who received the 1932 Nobel Prize in chemistry.

For his Ph.D., which Mac Lane received in 1934 from the Mathematisches Institut of Göttingen, Germany, he studied under two prominent mathematicians, Hermann Weyl and Paul Bernays.

Mac Lane worked as a mathematics instructor at Harvard and Cornell universities and at the University of Chicago from 1934 to 1938. He served again on the Harvard faculty from 1938 to 1947, and at Chicago beginning in 1947. In 1944 and 1945, he also directed Columbia University’s Applied Mathematics Group, which was involved in the war effort.

Mac Lane was Chairman of the Mathematics Department at Chicago from 1952 to 1958, taking the reins from Marshall Stone at a time many consider the high point in the department’s history. He was appointed the Max Mason Distinguished Service Professor in Mathematics in 1963, and became Professor Emeritus in 1982.

He was the author or co-author of more than 100 research papers and six books: A Survey of Modern Algebra (1941); Homology (1963); Algebra (1967); Categories for the Working Mathematician (1971); Mathematics, Form and Function (1985); and Sheaves in Geometry and Logic: A First Introduction to Topos Theory, with Ieke Moerdijk, (1992).

His A Survey of Modern Algebra, written with Garrett Birkhoff, was for years the leading textbook in its field and was republished by A K Peters Ltd. in 1997.

Mac Lane was elected to the National Academy of Sciences in 1949. He received the nation’s highest award for scientific achievement, the National Medal of Science, in 1989. Mac Lane received two Guggenheim Fellowships and visited Australia as a Fulbright Scholar. He also received honorary degrees from Purdue University, Yale University and the University of Glasgow, among others. Other honors include both the Chauvenet Prize and the Distinguished Service award of the Mathematical Association of America, the Steele Career Prize of the American Mathematical Society, and honorary fellowship in the Royal Society of Edinburgh.

The University of Chicago Alumni Association presented Mac Lane the Norman Maclean Faculty Award in 2003. The Maclean Award recognizes emeritus professors or senior faculty members who have made outstanding contributions to teaching and to the experience of student life on campus.

Mac Lane is survived by his widow, Osa, of San Francisco, and two daughters, Gretchen Mac Lane of New York City and Cynthia Hay of London; one grandson, William Hay, of London; three stepchildren, William Segal of Washington, D.C.; Andrew Segal of Santa Fe, N.M.; and Karen Segal of San Francisco;, and five step-grandchildren. Mac Lane’s first wife, Dorothy Jones Mac Lane, died in 1985.

Services for the family were held in San Francisco April 19. The Mathematical Sciences Research Institute in Berkeley will hold a memorial at 4:30 p.m. Wednesday, May 4.