Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who co-founded

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

with

Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...

Early life and education

Mac Lane was born in

Norwich, Connecticut Norwich ( ) (also called "The Rose of New England") is a city in New London County, Connecticut, United States. The Yantic, Shetucket, and Quinebaug Rivers flow into the city and form its harbor, from which the Thames River flows south to ...

, near where his family lived in Taftville.. He was christened "Leslie Saunders MacLane", but "Leslie" fell into disuse because his parents, Donald MacLane and Winifred Saunders, came to dislike it. He began inserting a space into his surname because his first wife found it difficult to type the name without a space. He was the oldest of three brothers; one of his brothers, Gerald MacLane, also became a mathematics professor at

Rice University William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranke ...

and

Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and ...

. Another sister died as a baby. His father and grandfather were both ministers; his grandfather had been a

Presbyterian Presbyterianism is a part of the Reformed tradition within Protestantism that broke from the Roman Catholic Church in Scotland by John Knox, who was a priest at St. Giles Cathedral (Church of Scotland). Presbyterian churches derive their n ...

, but was kicked out of the church for believing in

evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...

, and his father was a Congregationalist. His mother, Winifred, studied at

Mount Holyoke College Mount Holyoke College is a private liberal arts women's college in South Hadley, Massachusetts. It is the oldest member of the historic Seven Sisters colleges, a group of elite historically women's colleges in the Northeastern United States. ...

and taught English, Latin, and mathematics. In high school, Mac Lane's favorite subject was

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...

. While in high school, his father died, and he came under his grandfather's care. His half-uncle, a lawyer, was determined to send him to

Yale University Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the w ...

, where many of his relatives had been educated, and paid his way there beginning in 1926. As a freshman, he became disillusioned with chemistry. His mathematics instructor, Lester S. Hill, coached him for a local mathematics competition which he won, setting the direction for his future work. He went on to study mathematics and physics as a double major, taking courses from Jesse Beams,

Ernest William Brown Ernest William Brown FRS (29 November 1866 – 22 July 1938) was an English mathematician and astronomer, who spent the majority of his career working in the United States and became a naturalised American citizen in 1923. His life's work was ...

Ernest Lawrence Ernest Orlando Lawrence (August 8, 1901 – August 27, 1958) was an American nuclear physicist and winner of the Nobel Prize in Physics in 1939 for his invention of the cyclotron. He is known for his work on uranium-isotope separation fo ...

F. S. C. Northrop Filmer Stuart Cuckow Northrop (November 27, 1893 in Janesville, Wisconsin – July 22, 1992 in Exeter, New Hampshire) was an American legal philosopher and influential comparative philosopher. After receiving a B.A. from Beloit College in 19 ...

, and

Øystein Ore Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with ...

, among others. He graduated from Yale with a B.A. in 1930. During this period, he published his first scientific paper, in

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

and co-authored with

Irving Langmuir Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry. Langmuir's most famous publication is the 1919 ar ...

. In 1929, at a party of Yale football supporters in Montclair, New Jersey, Mac Lane (there to be presented with a prize for having the best grade point average yet recorded at Yale) had met Robert Maynard Hutchins, the new president of the

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

, who encouraged him to go there for his graduate studies and soon afterwards offered him a scholarship. Mac Lane neglected to actually apply to the program, but showed up and was admitted anyway. At Chicago, the subjects he studied included

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

with E. H. Moore,

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

with Leonard Eugene Dickson, the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

with Gilbert Ames Bliss, and logic with

Mortimer J. Adler Mortimer Jerome Adler (December 28, 1902 – June 28, 2001) was an American philosopher, educator, encyclopedist, and popular author. As a philosopher he worked within the Aristotelian and Thomistic traditions. He lived for long stretches in New ...

. In 1931, having earned his master's degree and feeling restless at Chicago, he earned a fellowship from the Institute of International Education and became one of the last Americans to study at the

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...

prior to its decline under the Nazis. His greatest influences there were

Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...

and

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...

. By the time he finished his doctorate in 1934, Bernays had been forced to leave because he was Jewish, and Weyl became his main examiner. At Göttingen, Mac Lane also studied with

Gustav Herglotz Gustav Herglotz (2 February 1881 – 22 March 1953) was a German Bohemian physicist best known for his works on the theory of relativity and seismology. Biography Gustav Ferdinand Joseph Wenzel Herglotz was born in Volary num. 28 to a public ...

and Emmy Noether. Within days of finishing his degree, he married Dorothy Jones, from Chicago, and soon returned to the U.S.

Career

From 1934 through 1938, Mac Lane held short-term appointments at

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...

Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...

, and the

. He then held a tenure track appointment at Harvard from 1938 to 1947. In 1941, while giving a series of visiting lectures at the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

, he met

and began what would become a fruitful collaboration on the interplay between algebra and topology. In 1944 and 1945, he also directed Columbia University's Applied Mathematics Group, which was involved in the war effort as a contractor for the Applied Mathematics Panel; the mathematics he worked on in this group concerned

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

s for

fire-control system A fire-control system (FCS) is a number of components working together, usually a gun data computer, a director, and radar, which is designed to assist a ranged weapon system to target, track, and hit a target. It performs the same task as a ...

s. In 1947, he accepted an offer to return to Chicago, where (in part because of the university's involvement in the

Manhattan Project The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...

, and in part because of the administrative efforts of

Marshall Stone Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras. Biography Stone was the son of Harlan Fiske Stone, who wa ...

) many other famous mathematicians and physicists had also recently moved. He traveled as a

Guggenheim Fellow Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

ETH Zurich (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , a ...

for the 1947–1948 term, where he worked with

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Eliza ...

. Mac Lane succeeded Stone as department chair in 1952, and served for six years. Mac Lane was vice president of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...

and the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

, and president of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...

. While presiding over the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

in the 1950s, he initiated its activities aimed at improving the teaching of modern mathematics. He was a member of the National Science Board, 1974–1980, advising the American government. In 1976, he led a delegation of mathematicians to

China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...

to study the conditions affecting mathematics there. Mac Lane was elected to the

in 1949, and received the National Medal of Science in 1989.

Contributions

After a thesis in

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

, his early work was in field theory and

valuation theory In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inhe ...

. He wrote on

valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' suc ...

s and Witt vectors, and separability in infinite

field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

s. He started writing on group extensions in 1942, and in 1943 began his research on what are now called Eilenberg–MacLane spaces K(''G'',''n''), having a single non-trivial

homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...

''G'' in dimension ''n''. This work opened the way to

group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...

in general. After introducing, via the

Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...

, the abstract approach to

homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

, he and Eilenberg originated

in 1945. He is especially known for his work on

coherence theorem In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax ...

s. A recurring feature of category theory,

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

, and of some other mathematics as well, is the use of diagrams, consisting of arrows (

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

s) linking objects, such as products and coproducts. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948). Mac Lane also coined the term

Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ...

for a lemma which is an essential background to many central concepts of category theory and which was discovered by

Nobuo Yoneda was a Japanese mathematician and computer scientist. In 1952, he graduated the Department of Mathematics, the Faculty of Science, the University of Tokyo, and obtained his Bachelor of Science. That same year, he was appointed Assistant Profess ...

. Mac Lane had an exemplary devotion to writing approachable texts, starting with his very influential ''A Survey of Modern Algebra'', coauthored in 1941 with Garrett Birkhoff. From then on, it was possible to teach elementary modern algebra to undergraduates using an English text. His ''

Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on ...

'' remains the definitive introduction to category theory. Mac Lane supervised the Ph.Ds of, among many others,

David Eisenbud David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously serve ...

, William Howard, Irving Kaplansky, Michael Morley,

Anil Nerode Anil Nerode (born 1932) is an American mathematician. He received his undergraduate education and a Ph.D. in mathematics from the University of Chicago, the latter under the directions of Saunders Mac Lane. He enrolled in the Hutchins College a ...

, Robert Solovay, and John G. Thompson. In addition to reviewing a fair amount of his mathematical output, the obituary articles McLarty (2005, 2007) clarify Mac Lane's contributions to the

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...

. Mac Lane (1986) is an approachable introduction to his views on this subject.

Selected works

* 1997 (1941). ''A Survey of Modern Algebra'' (with Garrett Birkhoff). A K Peters. *1948, "Groups, categories and duality," ''Proceedings of the Nat. Acad. of Sciences of the USA 34'': 263–67. * 1995 (1963). ''Homology'', Springer ( Classics in Mathematics) (Originally, Band 114 of Die Grundlehren Der Mathematischen Wissenschaften in Einzeldarstellungen.
AMS review
by

David Buchsbaum David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University who worked on commutative algebra, homological algebra, and representation theory. He proved the Auslander–Buchsbaum formula and the Auslan ...

. * 1999 (1967). ''Algebra'' (with Garrett Birkhoff). Chelsea. * 1998 (1972). ''

'', Springer (

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...

) * 1986. ''

Mathematics, Form and Function ''Mathematics, Form and Function'', a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane. Mathematics and human activities ...

''. Springer-Verlag. * 1992. '' Sheaves in Geometry and Logic: A First Introduction to Topos Theory'' (with

Ieke Moerdijk Izak (Ieke) Moerdijk (; born 23 January 1958) is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize. Education and career Moerdijk studied mathematics, philosophy and general linguistics at the Un ...

). * 1995. * 2005. ''Saunders Mac Lane: A Mathematical Autobiography''. A K Peters.

Notes

References

* (

e-book An ebook (short for electronic book), also known as an e-book or eBook, is a book publication made available in digital form, consisting of text, images, or both, readable on the flat-panel display of computers or other electronic devices. A ...

: ).

Biographical references

*. With selected bibliography emphasizing Mac Lane's philosophical writings. *. * .

External links

*
Obituary press release
from the

.
Photographs of Mac Lane
1984–1999. * Kutateladze S.S.
Saunders Mac Lane, the Knight of Mathematics
* {{DEFAULTSORT:Mac Lane, Saunders 1909 births 2005 deaths Mathematicians from Connecticut People from Norwich, Connecticut 20th-century American mathematicians 21st-century American mathematicians University of Chicago alumni University of Göttingen alumni Yale University alumni Algebraists Category theorists Cornell University faculty Harvard University faculty University of Chicago faculty National Medal of Science laureates Members of the American Philosophical Society Members of the United States National Academy of Sciences Presidents of the American Mathematical Society Presidents of the Mathematical Association of America Proceedings of the National Academy of Sciences of the United States of America editors