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 für Mathematik at the ETH in Zurich 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. 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 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. 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.
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