William Lawvere

Born in Muncie, Indiana, and raised on a farm outside Mathews, Lawvere received his undergraduate degree in mathematics from Indiana University.^[1]

Lawvere studied continuum mechanics and kinetic theory as an undergraduate with Clifford Truesdell.^[2] 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.

Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. Truesdell supported Lawvere's application to study further with Samuel Eilenberg, a founder of category theory, at Columbia University in 1960.^[1][3]

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 an ingredient (the constant case) of elementary topos theory.

Lawvere died on January 23, 2023, in Chapel Hill, N.C., after a long illness at the age of 85.^[1][3]

Lawvere completed his Ph.D. at Columbia in 1963 with Eilenberg. His dissertation introduced the category of categories as a framework for the semantics of algebraic theories. From 1964 to 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'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. Lawvere's 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 Zürich for 1968 and 1969 he proposed elementary (first-order) axioms for toposes generalizing the concept of the Grothendieck topos (see History of topos theory) and worked with the algebraic topologist Myles Tierney to clarify and apply 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.^[4] 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.^{[needs context]} 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 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 continued to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications. He was professor emeritus of mathematics and adjunct professor emeritus of philosophy at Buffalo.^[3]

A central motivation for Lawvere's work is the search for a good mathematical (rigorous) foundations of physics, specifically of (classical) continuum mechanics (or at least some kinematical aspects thereof, Lawvere does not seem to mention Hamiltonians, Lagrangians or action functionals).^[5]

In an interview (page 8) he recalled:^[2]

I had been a student at Indiana University from 1955 to January 1960. I liked experimental physics but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study mathematics first. Truesdell was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there. ... in 1955 (and subsequently) had advised me on pursuing the study of continuum mechanics and kinetic theory. In Summer 1958, I studied Topological Dynamics with George Whaples, with the agenda of understanding as much as possible in categorical terms. ... Categories would clearly be important for simplifying the foundations of continuum physics. I concluded that I would make category theory a central line of my study.

Then in the same interview (page 11) he said about the early 1960s:

I felt a strong need to learn more set theory and logic from experts in that field, still of course with the aim of clarifying the foundations of category theory and of physics.

The title of the early text "Toposes of laws of motion", which is often cited as the text introducing synthetic differential geometry, clearly witnesses the origin and motivation of these ideas in classical mechanics.^[6]

In an interview, William F. Lawvere reflects on his time as an assistant professor at the University of Chicago in 1967. He mentions that he and Mac Lane co-taught a course on mechanics, which led him to consider the justification of older intuitive methods in geometry, eventually coining the term "synthetic differential geometry" This course was based on Mackey's book Mathematical Foundations of Quantum Mechanics, indicating Mackey's influence on category theory.^[2]

Further in the interview, he discusses the origins of synthetic differential geometry, noting that the idea for the joint course on mechanics came from a suggestion by Chandra. This course was the first in a series, and Mac Lane later gave a talk on the Hamilton-Jacobi equation at the Naval Academy in 1970, which was published in The American Mathematical Monthly. He explains that he began applying Grothendieck topos theory, learned from Gabriel, to simplify the foundations of continuum mechanics, inspired by Truesdell's teachings, Noll's axiomatizations, and his own efforts in 1958 to categorize topological dynamics.

A more detailed review of these ideas and their relation to physics can be found in the introduction to the book collection Categories in Continuum Physics, which is the proceedings of a meeting organized by Lawvere in 1982.^[7]

In his 1997 talk "Toposes of Laws of Motion", Lawvere remarks on the longstanding program of infinitesimal calculus, continuum mechanics, and differential geometry, which aims to reconstruct the world from the infinitely small. He acknowledges the skepticism around this idea but emphasizes its fruitful outcomes over the past 300 years. He believes that recent developments have positioned mathematicians to make this program more explicit, focusing on how continuum physics can be mathematically constructed from "simple ingredients".^[6]

In the same talk, Lawvere mentions that the essential spaces required for functional analysis and physical field theories can be found in any topos with an appropriate object (T).

In his 2000 article "Comments on the Development of Topos Theory", Lawvere discusses his motivation for simplifying and generalizing Grothendieck's concept of topos. He explains that his interest stemmed from his earlier studies in physics, particularly the foundations of continuum physics as inspired by Truesdell, Noll, and others. He notes that while the mathematical apparatus used in this field is powerful, it often does not fit the phenomena well. Lawvere questions whether the problems and necessary axioms could be stated more directly and clearly, potentially leading to a simpler yet rigorous account. These questions led him to apply the topos method in his 1967 Chicago lectures on categorical dynamics. He realized that further work on the notion of topos was necessary to achieve his goals. His time spent with Berkeley logicians in 1961-62, listening to experts on foundations, also influenced his approach.^[8]

Lawvere highlights that several books on simplified topos theory, including the recent and accessible text by MacLane and Moerdijk, along with three excellent books on synthetic differential geometry, provide a solid foundation for further work in functional analysis and the development of continuum physics.

William Lawvere has also proposed formalizations in category theory, categorical logic and topos theory of concepts which are motivated from philosophy, notably in Georg Hegel's Science of Logic (see there for more).

This includes for instance definitions of concepts found there such as objective and subjective logic, abstract general, concrete general, concrete particular, unity of opposites, Aufhebung, being, becoming, space and quantity, cohesion, intensive and extensive quantity ... and so on.^[5]

In his work "Categories of Space and Quantity" from The Space of Mathematics (1992), William Lawvere expresses his belief that the technical advancements made by category theorists will significantly benefit dialectical philosophy in the coming decades and century. He argues that these advancements will provide precise mathematical models for age-old philosophical distinctions, such as general versus particular, objective versus subjective, and being versus becoming. He emphasizes that mathematicians need to engage with these philosophical questions to make mathematics and other sciences more accessible and useful. This, he notes, will require philosophers to learn mathematics and mathematicians to learn philosophy.^[9]

A precursor to this undertaking is Hermann Grassmann with his Ausdehnungslehre.^[10]

The category theorist William Lawvere was a committed Marxist-Leninist; at one point he gave a talk called "Applying Marxism-Leninism-Mao Tse-Thung Thought to Mathematics & Science".

According to Anders Kock's obituary, in 1971:^[11]

the [Dalhousie] university administration refused to renew the contract with [Lawvere], due to his political activities in protesting against the Vietnam war and against the War Measures Act proclaimed by Trudeau, suspending civil liberties under the pretext of danger of terrorism.

As per the obituary on the Communist Party of Canada (Marxist–Leninist) site:^[12]

More than 1,000 students rallied in the lobby of the Dal Student Union Building to oppose the arbitrary dismissal of Professor Lawvere.

He saw his political commitments as related to his mathematical work in sometimes surprising and unexpected ways: for instance, here's a passage from Quantifiers and Sheaves (1970):^[13]

When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing. Doing this for "set theory" requires taking account of the experience that the main pairs of opposing tendencies in mathematics take the form of adjoint functors, and frees us of the mathematically irrelevant traces (∈) left behind by the process of accumulating (∪) the power set (P) at each stage of a metaphysical "construction".

In the earlier sections of the paper, he discusses the "unity of opposites" between logic and geometry. He clarifies that his discussion of contradiction, ideology, and opposition is rooted in the Marxist tradition, referencing Mao's "On Contradiction" (1937) in the bibliography. Additionally, he connects various mathematical concepts to Hegel's Dialectic and Lenin's theory of knowledge in other parts of his work.

• In 2010 he received the "Premio Giulio Preti", awarded by the Regional Council of Tuscany
• In 2012 he became a fellow of the American Mathematical Society.^[14]

• 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. ISBN 3-540-16096-5; ebook
• 2003 (2002) Sets for Mathematics (with Robert Rosebrugh). Cambridge Uni. Press. ISBN 0-521-01060-8
• 2009 Conceptual Mathematics: A First Introduction to Categories (with Stephen H. Schanuel). Cambridge University Press, 2nd ed. ISBN 978-0521719162; 1997 pbk edition

• Category theory
• Topos theory
• History of topos theory
• Synthetic differential geometry
• Philosophy of mathematics
• Dialectic
• Marxism–Leninism
• Lawvere–Tierney topology

• 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)
• Reprints in Theory and Applications of Categories. 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.
• William F. Lawvere nLab entry
• Repository of William F. Lawvere's collected works hosted on GitHub.
• William Lawvere at the Mathematics Genealogy Project
• Photograph