William Lawvere
Encyclopedia
Francis William Lawvere (born February 9, 1937) is a mathematician
known for his work in category theory
, topos theory
and the philosophy of mathematics
.
as an undergraduate with Clifford Truesdell
. He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley
's textbook General Topology where Kelley suggests the functorial approach "might be called the galactic theory" (p. 246) compared to the older ideas of local and global questions. Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll
. Truesdell, who had an appointment in mathematics himself, supported Lawvere's application to study more pure mathematics
with Samuel Eilenberg
, a founder of category theory, at Columbia University
in 1960.
Before completing the Ph.D. Lawvere spent a year in Berkeley
as an informal student of model theory
and set theory
, following lectures by Alfred Tarski
and Dana Scott
. In his first teaching position at Reed College
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) which became a key ingredient (the constant case) of elementary topos theory
.
's seminars at Oberwolfach
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. His Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advanced categorical logic
especially using his 1963 discovery that existential and universal quantifiers can be characterized as special cases of adjoint functors
.
Back in Zurich for 1968-69 he proposed elementary (first-order) axioms for toposes generalizing the concept of the Grothendieck
topos (see background and genesis of topos theory
) and worked with the algebraic topologist Myles Tierney to clarifying and applying this theory. Tierney discovered major simplifications in the description of Grothendieck "topologies". Anders Kock 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
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
, 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
, often collaborating with Stephen Schanuel
. In 1977 he was elected to the Martin professorship in mathematics for 5 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
which 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 of mathematics and adjunct professor emeritus of philosophy at Buffalo.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for his work in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, topos theory
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
and the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
.
Biography
Lawvere studied continuum mechanicsContinuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
as an undergraduate with Clifford Truesdell
Clifford Truesdell
Clifford Ambrose Truesdell III was an American mathematician, natural philosopher, historian of science, and polemicist.-Life:...
. He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley
John L. Kelley
John Leroy Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis....
's textbook General Topology where Kelley suggests the functorial approach "might be called the galactic theory" (p. 246) compared to the older ideas of local and global questions. Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll
Walter Noll
Walter Noll is a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics and thermodynamics....
. Truesdell, who had an appointment in mathematics himself, supported Lawvere's application to study more pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
with Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
, a founder of category theory, at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
in 1960.
Before completing the Ph.D. Lawvere spent a year in Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
as an informal student of model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, following lectures by Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
and Dana Scott
Dana Scott
Dana Stewart Scott 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, California...
. In his first teaching position at Reed College
Reed College
Reed College is a private, independent, liberal arts college located in southeast Portland, Oregon. Founded in 1908, Reed is a residential college with a campus located in Portland's Eastmoreland neighborhood, featuring architecture based on the Tudor-Gothic style, and a forested canyon wilderness...
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) which became a key ingredient (the constant case) of elementary topos theory
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
.
Work
Lawvere completed his Ph.D at Columbia in 1963 with Eilenberg. His dissertation introduced the Category of Categories in his thesis as a framework for the semantics of algebraic theories. During 1964-1967 at the Forschungsinstitut fuer Mathematik at the ETH in Zurich he worked on the Category of Categories and was especially influenced by Pierre GabrielPierre Gabriel
Pierre Gabriel is a mathematician at Universität Zürich who works on category theory, algebraic groups, and representation theory of algebras. He was elected a correspondent member of the French Academy of Sciences in November 1986.-See also:...
's seminars at Oberwolfach
Oberwolfach
Oberwolfach is a town in the district of Ortenau in Baden-Württemberg, Germany. It is the site of the Mathematical Research Institute of Oberwolfach, or Mathematisches Forschungsinstitut Oberwolfach.- Geographical situation :...
on Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
's foundation of algebraic geometry. He then taught at the University of Chicago, working with Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
, and at the City University of New York Graduate Center (CUNY), working with Alex Heller. His Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advanced categorical logic
Categorical logic
Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor...
especially using his 1963 discovery that existential and universal quantifiers can be characterized as special cases of adjoint functors
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
.
Back in Zurich for 1968-69 he proposed elementary (first-order) axioms for toposes generalizing the concept of the Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
topos (see background and genesis of topos theory
Background and genesis of topos theory
This page 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 other hand context can be given...
) and worked with the algebraic topologist Myles Tierney to clarifying and applying this theory. Tierney discovered major simplifications in the description of Grothendieck "topologies". Anders Kock 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
Dalhousie University is a public research university located in Halifax, Nova Scotia, Canada. The university comprises eleven faculties including Schulich School of Law and Dalhousie University Faculty of Medicine. It also includes the faculties of architecture, planning and engineering located at...
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
War Measures Act
The War Measures Act was a Canadian statute that allowed the government to assume sweeping emergency powers in the event of "war, invasion or insurrection, real or apprehended"...
, 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
University at Buffalo, The State University of New York
University at Buffalo, The State University of New York, also commonly known as the University at Buffalo or UB, is a public research university and a "University Center" in the State University of New York system. The university was founded by Millard Fillmore in 1846. UB has multiple campuses...
, often collaborating with Stephen Schanuel
Stephen Schanuel
Stephen H. Schanuel is an American mathematician working in the fields of abstract algebra and number theory, more specifically category theory and measure theory....
. In 1977 he was elected to the Martin professorship in mathematics for 5 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
Synthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
which 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 of mathematics and adjunct professor emeritus of philosophy at Buffalo.
Selected books
- 1986 Categories in Continuum Physics (Buffalo, N.Y. 1982), edited by Lawvere and Stephen H. SchanuelStephen SchanuelStephen H. Schanuel is an American mathematician working in the fields of abstract algebra and number theory, more specifically category theory and measure theory....
(with Introduction by Lawvere pp 1–16), Springer Lecture Notes in Mathematics 1174. ISBN 3-540-16096-5 - 1997 Conceptual Mathematics: A First Introduction to Categories (with Stephen H. Schanuel). Cambridge Uni. Press. ISBN 0-521-47817-0
- 2003 (2002) Sets for Mathematics (with Robert Rosebrugh). Cambridge Uni. Press. ISBN 0-521-01060-8
External links
- A recent interview published on the Bulletin of the International Center for Mathematics of Coimbra, Portugal (Part I , Part II)
- http://www.tac.mta.ca/tac/reprints/index.html Includes reprints of seven 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.
- Photograph
- John Baez's This Week's Finds in Mathematical Physics (Week 200)