William Lawvere

Francis William Lawvere (/lɔːˈvɪər/; February 9, 1937 – January 23, 2023) was an American mathematician and philosopher known for his foundational work in category theory, topos theory, and the philosophy of mathematics. He introduced algebraic theories as categories, developed the Elementary Theory of the Category of Sets (ETCS) as an alternative foundation of mathematics, founded categorical logic and co-founded the field of topos theory. A central theme of his work was the search for a rigorous mathematical foundation for classical continuum mechanics and physics based on categorical methods.

Born in Muncie, Indiana, and raised on a farm outside Mathews, Lawvere began his university studies at Indiana University in 1955. There, he studied continuum mechanics with Clifford Truesdell and philosophy with Alan Donagan. He learned of category theory while preparing to teach 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 creating simple, rigorous axioms for the physical ideas of Truesdell and Walter Noll. Truesdell supported Lawvere's application to study with Samuel Eilenberg, a founder of category theory, at Columbia University in 1960.

Before completing his Ph.D., Lawvere spent the 1961–62 academic year at the University of California, 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 developed the first axioms for the composition of mappings, which later evolved into the Elementary Theory of the Category of Sets (1964).

Lawvere completed his Ph.D. at Columbia in 1963 with Eilenberg. From 1964 to 1967, at the Forschungsinstitut für Mathematik at the ETH in Zürich, he worked on the category of categories and was influenced by Pierre Gabriel's seminars at Oberwolfach on Grothendieck's foundations of algebraic geometry. He then taught at the University of Chicago (1967-68), working with Mac Lane, and at the CUNY Graduate Center (1968-69), working with Alex Heller. He returned to the ETH from 1968 to 1969.

In 1969, Dalhousie University established a research group of 15 researchers with Lawvere at its head, supported by a Killam grant. However, the university terminated the group and dismissed Lawvere in 1971, citing his political activities and for teaching the history of mathematics without permission. The dismissal sparked a significant protest from the student body. Despite the controversy, in 1995 Dalhousie hosted a celebration of 50 years of category theory with both Lawvere and Saunders Mac Lane present.

After leaving Dalhousie, Lawvere ran a seminar in Perugia, Italy (1972–1974), where he worked on enriched categories. In 1974, he became a professor of mathematics at the University at Buffalo, where he remained until his retirement in 2000, often collaborating with Stephen Schanuel. In 1977, he was elected to the Martin professorship in mathematics for five years, which made possible the 1982 meeting on "Categories in Continuum Physics." He was professor emeritus of mathematics and adjunct professor emeritus of philosophy at Buffalo.

Lawvere died on January 23, 2023, in Chapel Hill, North Carolina, after a long illness at the age of 85.

Lawvere's work is characterized by the use of category theory to provide simple, general foundations for mathematical and physical concepts.