Francis William Lawvere (; born February 9, 1937) is a
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 ...
known for his work in
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, cate ...
,
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 ...
and 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's ...
.
Biography
Lawvere studied
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
as an undergraduate with
Clifford Truesdell. He learned of category theory while teaching a course on
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
for Truesdell, specifically from a problem in
John L. Kelley's textbook ''General Topology''. Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and
Walter Noll
Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics.
Biography ...
. Truesdell supported Lawvere's application to study further 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 a ...
, a founder of category theory, at
Columbia University
Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...
in 1960.
Before completing the Ph.D. Lawvere spent a year in
Berkeley
Berkeley most often refers to:
*Berkeley, California, a city in the United States
**University of California, Berkeley, a public university in Berkeley, California
* George Berkeley (1685–1753), Anglo-Irish philosopher
Berkeley may also refer ...
as an informal student of
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
and
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 conce ...
, following lectures by
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
and
Dana Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...
. In his first teaching position at
Reed College
Reed College is a private liberal arts college in Portland, Oregon. Founded in 1908, Reed is a residential college with a campus in the Eastmoreland neighborhood, with Tudor-Gothic style architecture, and a forested canyon nature preserve at ...
he was instructed to devise courses in calculus and abstract algebra from a foundational perspective. He tried to use the then current axiomatic set theory but found it unworkable for undergraduates, so he instead developed the first axioms for the more relevant composition of mappings of sets. He later streamlined those axioms into the ''Elementary Theory of the Category of Sets'' (1964)
(Reprints, #11), which became an ingredient (the constant case) of elementary
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 ...
.
Work
Lawvere completed his
Ph.D
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common Academic degree, degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields ...
at Columbia in 1963 with Eilenberg. His dissertation introduced the Category of Categories as a framework for the semantics of
algebraic theories. During 1964–1967 at the Forschungsinstitut für Mathematik at the ETH in Zürich he worked on the Category of Categories and was especially influenced by
Pierre Gabriel
Pierre Gabriel (1 August 1933 – 24 November 2015), also known as Peter Gabriel, was a French mathematician at the University of Strasbourg (1962–1970), University of Bonn (1970–1974) and University of Zürich (1974–1998) who worked on cat ...
's seminars at
Oberwolfach
Oberwolfach ( gsw, label= Low Alemannic, Obberwolfä) is a town in the district of Ortenau in Baden-Württemberg, Germany. It is the site of the Oberwolfach Research Institute for Mathematics, or Mathematisches Forschungsinstitut Oberwolfach.
Ge ...
on
Grothendieck's foundation of algebraic geometry. He then taught at the University of Chicago, working with
Mac Lane, and at the City University of New York Graduate Center (CUNY), working with
Alex Heller
Alex is a given name. It can refer to a shortened version of Alexander, Alexandra, Alexis.
People
Multiple
*Alex Brown (disambiguation), multiple people
* Alex Gordon (disambiguation), multiple people
*Alex Harris (disambiguation), multiple peo ...
. His Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advanced
categorical logic
__NOTOC__
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science.
In broad terms, categ ...
especially using his 1963 discovery that existential and universal
quantifiers can be characterized as special cases of
adjoint functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
.
Back in Zürich for 1968–69 he proposed elementary (first-order) axioms for toposes generalizing the concept of the
Grothendieck topos (see
history of topos theory
This article gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the ...
) and worked with the algebraic topologist
Myles Tierney
Myles Tierney (September 1937 – 5 October 2017) was an American mathematician and Professor at Rutgers University who founded the theory of elementary toposes with William Lawvere.
Tierney obtained his B.A. from Brown University in 1959 and his ...
to clarify and apply this theory. Tierney discovered major simplifications in the description of Grothendieck "topologies".
Anders Kock
Anders is a male name in Scandinavian languages and Fering North Frisian, an equivalent of the Greek Andreas ("manly") and the English Andrew. It originated from Andres via metathesis.
In Sweden, Anders has been one of the most common names fo ...
later found further simplifications so that a topos can be described as a category with products and equalizers in which the notions of map space and subobject are representable. Lawvere had pointed out that a Grothendieck topology can be entirely described as an endomorphism of the subobject representor, and Tierney showed that the conditions it needs to satisfy are just idempotence and the preservation of finite intersections. These "topologies" are important in both algebraic geometry and model theory because they determine the subtoposes as sheaf-categories.
Dalhousie University
Dalhousie University (commonly known as Dal) is a large public research university in Nova Scotia
Nova Scotia ( ; ; ) is one of the thirteen provinces and territories of Canada. It is one of the three Maritime provinces and one of the fou ...
in 1969 set up a group of 15 Killam-supported researchers with Lawvere at the head; but in 1971 it terminated the group. Lawvere was controversial for his political opinions, for example, his opposition to the 1970 use of the
War Measures Act
The ''War Measures Act'' (french: Loi sur les mesures de guerre; 5 George V, Chap. 2) was a statute of the Parliament of Canada that provided for the declaration of war, invasion, or insurrection, and the types of emergency measures that could t ...
, and for teaching the history of mathematics without permission. But in 1995 Dalhousie hosted the celebration of 50 years of category theory with Lawvere and Saunders Mac Lane present.
Lawvere ran a seminar in Perugia, Italy (1972–1974) and especially worked on various kinds of enriched category. For example, a metric space can be regarded as an enriched category. From 1974 until his retirement in 2000 he was professor of mathematics at
University at Buffalo
The State University of New York at Buffalo, commonly called the University at Buffalo (UB) and sometimes called SUNY Buffalo, is a public research university with campuses in Buffalo and Amherst, New York. The university was founded in 1846 ...
, often collaborating with
Stephen Schanuel
Stephen H. Schanuel (14 July 1933 – 21 July 2014) was an American mathematician working in the fields of abstract algebra and category theory, number theory, and measure theory.
Life
While he was a graduate student at University of Chicago, ...
. In 1977 he was elected to the Martin professorship in mathematics for five years, which made possible the meeting on "Categories in Continuum Physics" in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the
synthetic differential geometry that had evolved from the spatial part of Lawvere's categorical dynamics program. Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications. He is now
professor emeritus
''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...
of mathematics and adjunct professor emeritus of philosophy at Buffalo.
Awards and honors
* In 2010 he received the "Premio Giulio Preti", awarded by the Consiglio regionale della Toscana
* In 2012 he became a fellow 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, ...
.
List of Fellows of the American Mathematical Society
retrieved 2013-01-27.
Selected books
* 1986 ''Categories in Continuum Physics'' (Buffalo, N.Y. 1982), edited by Lawvere and Stephen H. Schanuel (with Introduction by Lawvere pp 1–16), Springer Lecture Notes in Mathematics 1174.
ebook
* 2003 (2002) ''Sets for Mathematics'' (with Robert Rosebrugh). Cambridge Uni. Press.
* 2009 ''Conceptual Mathematics: A First Introduction to Categories'' (with Stephen H. Schanuel). Cambridge University Press, 2nd ed.
1997 pbk edition
See also
*Lawvere–Tierney topology
In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology o ...
References
External links
* A 2007 interview published on the Bulletin of the International Center for Mathematics of Coimbra, Portugal
Part I
Part II
both parts in one file
Includes reprints of eight of Lawvere's fundamental articles, among them his dissertation and his first full treatment of the category of sets. Those two had circulated only as mimeographs.
Homepage.
Includes bibliography and downloadable papers, Ph.D. thesis.
*
{{DEFAULTSORT:Lawvere, William
1937 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Category theorists
University at Buffalo faculty
University of Chicago faculty
Columbia University alumni
Fellows of the American Mathematical Society
Philosophers of mathematics
20th-century American philosophers
21st-century American philosophers