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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.
In mathematics, Lawvere's fixed-point theorem is an important result in category theory. [1] It is a broad abstract generalization of many diagonal arguments in mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem.
Formally, a Lawvere theory consists of a small category L with (strictly associative) finite products and a strict identity-on-objects functor: preserving finite products. A model of a Lawvere theory in a category C with finite products is a finite-product preserving functor M : L → C .
(PDF). Notices of the AMS. 51 (9): 160– 1. The following texts are easy-paced introductions to toposes and the basics of category theory. They should be suitable for those knowing little mathematical logic and set theory, even non-mathematicians. Lawvere, F. William; Schanuel, Stephen H. (1997). Conceptual Mathematics: A First Introduction to ...
This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b.In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT (ε ≠ 0); yet it is provably false that all infinitesimals are equal to zero. [2]
Each kind of category with extra structure corresponds to a system of logic with its own inference rules. A Lawvere theory is an algebraic theory as a category with finite products and possessing a "generic algebra" (a generic group). The structures described by a Lawvere theory are models of the Lawvere theory 1963: Jean-Louis Verdier