Francis William Lawvere (1937-) is an American mathematician working in category theory. He received his Ph.D. from Columbia University in 1963 under the supervision of Samuel Eilenberg. He was a professor at Buffalo University NY from 1974 until his retirement in 2000. His students include Marta Bunge? and Anders Kock?.
In his thesis, Functorial Semantics of Algebraic Theories, Lawvere gives a categorical formulation to universal algebra, introducing the functorial viewpoint in model theory. He also proposes a foundation for mathematics based on the Category of Categories. During the same period, he showed that the rules of predicate logic can be deduced from properties of adjoint functors. He later introduced a general notion of logical theory called Hyperdoctrine, in which the logical connectives where replaced by more general categorical operations. His Elementary Theory of the Category of Sets (1964) was a precursor of the notion of Elementary Topos which he later invented in collaboration with Myles Tierney? in 1969-1971. The notion was obtained by regarding a Grothendieck topos as a category equipped with certain basic operations, similar to those of the Elementary Theory of the Category of Sets. The Lawvere object , which classifies subobjects in a topos, is playing a central role in elementary topos theory. The object was actually discovered earlier by the Grothendieck school?, but its importance was only recognised by Lawvere. It turns out that every elementary topos carries an internal logic which is intuitionistic in general. Toposes (or topoi) can be used for constructing new models of set theory, intuitionistic or classical. See topos in wikipedia. Many independance results of set theory were reproved using sheaf theory, for example, the independance of the continuum hypothesis by Tierney?. Peter Johnstone has published in 2004 two volumes on topos theory, Sketches of an elephant, based on the Lawvere-Tierney axiomatic approach.
Lawvere is regarded by many as the main leader of category theory after Eilenberg and Mac Lane. He is famous for questioning the current set theoretic foundation of mathematics. He has advocated that mathematics should benefit from an all categorical formulation. He also saw a connection between the materialism of Marx and Engels and the modern development of category theory. He was an active Marxist-Leninist during the 1970’s.