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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.
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 .
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.
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]
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
Fairly accessible introduction, but somewhat dated. The categorical approach to higher-order logics over polymorphic and dependent types was developed largely after this book was published. Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics. Vol. 141. North Holland, Elsevier. ISBN 0-444 ...