Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded

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

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 ( ) is a city in New London County, Connecticut, United States. The Yantic River, Yantic, Shetucket River, Shetucket, and Quinebaug Rivers flow into the city and form its harbor, from which the Thames River (Connecticut), Thames River f ...

, 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 eldest of three brothers; one of his brothers, Gerald MacLane, also became a mathematics professor at

Rice University William Marsh Rice University, commonly referred to as Rice University, is a Private university, private research university in Houston, Houston, Texas, United States. Established in 1912, the university spans 300 acres. Rice University comp ...

and

Purdue University Purdue University is a Public university#United States, public Land-grant university, land-grant research university in West Lafayette, Indiana, United States, and the flagship campus of the Purdue University system. The university was founded ...

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

Presbyterian Presbyterianism is a historically Reformed Protestant tradition named for its form of church government by representative assemblies of elders, known as "presbyters". Though other Reformed churches are structurally similar, the word ''Pr ...

, but was kicked out of the church for believing in

evolution Evolution is the change in the heritable Phenotypic trait, characteristics of biological populations over successive generations. It occurs when evolutionary processes such as natural selection and genetic drift act on genetic variation, re ...

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

Mount Holyoke College Mount Holyoke College is a Private college, private Women's colleges in the United States, women's Liberal arts colleges in the United States, liberal arts college in South Hadley, Massachusetts, United States. It is the oldest member of the h ...

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 physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

. 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 university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...

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

, F. S. C. Northrop, and Øystein Ore, 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 scientific study of matter, its Elementary particle, 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 whi ...

and co-authored with

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

. In 1929, at a party of Yale football supporters in

Montclair, New Jersey Montclair is a Township (New Jersey), township in Essex County, New Jersey, Essex County in the U.S. state of New Jersey. Situated on the cliffs of the Watchung Mountains, Montclair is a commercial and cultural hub of North Jersey and a diverse ...

, 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 Robert Maynard Hutchins (January 17, 1899 – May 14, 1977) was an American educational philosopher. He was the President of the University of Chicago, 5th president (1929–1945) and chancellor (1945–1951) of the University of Chicago, and ear ...

, the new president of the

University of Chicago The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...

, 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 Set (mathematics), 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 mathema ...

with E. H. Moore,

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

with

Leonard Eugene Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also rem ...

, 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 Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

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, popular author and lay theologian. As a philosopher he worked within the Aristotelian and Thomistic traditions. He taught at ...

. 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 (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...

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 ...

and

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

. 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 n ...

and

Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...

. 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 university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...

, 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 The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

, he met

and began what would become a fruitful collaboration on the interplay between algebra and topology. In 1944 and 1945, he 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 equations 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 hum ...

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 program undertaken during World War II to produce the first nuclear weapons. It was led by the United States in collaboration with the United Kingdom and Canada. From 1942 to 1946, the ...

, and in part because of the administrative efforts of Marshall Stone) 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, endowed by the late Simon and Olga Hirsh Guggenheim. These awards are bestowed upon individuals who have demonstrated d ...

ETH Zurich ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ran ...

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 dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...

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

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, 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 ...

and the

American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...

, 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, meetings, ...

. 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 A university () is an educational institution, institution of tertiary edu ...

in the 1950s, he initiated its activities aimed at improving the teaching of modern mathematics. He was a member of the

National Science Board The National Science Board (NSB) of the United States establishes the policies of the National Science Foundation (NSF) within the framework of applicable national policies set forth by the President of the United States, president and the United ...

, 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. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...

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

in 1949, and received the

National Medal of Science The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral science, behavior ...

in 1989.

Contributions

After a thesis in

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, Mac Lane's early work was in field theory and valuation theory. He wrote on

valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every non-zero 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 ' ...

s and

Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field o ...

s, and separability in infinite

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

s. He started writing on

group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...

s in 1942, and in 1943 began his research on what are now called

Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...

s 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, homo ...

''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 homolo ...

, the abstract approach to

homology theory In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...

, he and Eilenberg originated

in 1945. He is especially known for his work on coherence theorems. A recurring feature of category theory,

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, and of some other mathematics as well, is the use of

diagrams A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...

, consisting of arrows (

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s) linking objects, such as

products Product may refer to: Business * Product (business), an item that can be offered to a market to satisfy the desire or need of a customer. * Product (project management), a deliverable or set of deliverables that contribute to a business solution ...

and

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

s. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948), who also coined the term

Yoneda lemma In mathematics, the Yoneda lemma is a fundamental 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 (viewing a group as a ...

for a lemma which is an essential background to many central concepts of category theory and which was discovered by Nobuo Yoneda. 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 Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician Ge ...

. 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 ...

'' remains the definitive introduction to category theory.

Selected works

* 1997 (1941). ''A Survey of Modern Algebra'' (with

). A K Peters. *1948, "Groups, categories and duality," ''Proceedings of the Nat. Acad. of Sciences of the USA 34'': 263–67. * 1963. * 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 Ausland ...

. * 1999 (1967). ''Algebra'' (with

). Chelsea. * 1998 (1972). ''

'', Springer (

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () 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 size (with va ...

) * 1986. '' Mathematics, Form and Function''. Springer-Verlag. * 1992. '' Sheaves in Geometry and Logic: A First Introduction to

Topos Theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion ...

'' (with Ieke Moerdijk). * 1995. * 2005. ''Saunders Mac Lane: A Mathematical Autobiography''. A K Peters.

Notes

References

* (

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

: ).

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 American algebraists Category theorists Columbia University faculty Cornell University faculty Harvard University Department of Mathematics 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