[2503.13536] A Survey on Lawvere's Fixed-Point Theorem

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Mathematics > General Mathematics

arXiv:2503.13536 (math)

[Submitted on 15 Mar 2025]

Title:A Survey on Lawvere's Fixed-Point Theorem

Authors:Joaquim Reizi Barreto<br>View a PDF of the paper titled A Survey on Lawvere's Fixed-Point Theorem, by Joaquim Reizi Barreto

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Abstract:This paper provides an overview of Lawvere's Fixed-Point Theorem in category theory and aims to detail the universal framework underlying self-reference and recursive structures. First, we rigorously define fundamental concepts - such as terminal objects, products, Cartesian Closed Categories, exponential objects, evaluation maps, currying, and point-surjective morphisms - and explain their intuitive meanings through concrete examples and commutative diagrams. Based on these foundational notions, we derive key lemmas (the universality of currying, the diagonal lemma, and the fixed-point construction lemma) and integrate them to develop a proof of Lawvere's Fixed-Point Theorem. Furthermore, we discuss the impact of this theorem on fixed-point combinators in programming languages, type theory, and homotopy type theory, as well as current research trends and open problems. In doing so, we clarify how the abstract principle of self-reference contributes to a wide range of applications in both mathematics and computational theory.

Comments:<br>29 pages, 1 figure, and supplementary appendices

Subjects:

General Mathematics (math.GM)

Cite as:<br>arXiv:2503.13536 [math.GM]

(or<br>arXiv:2503.13536v1 [math.GM] for this version)

https://doi.org/10.48550/arXiv.2503.13536

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arXiv-issued DOI via DataCite

Submission history<br>From: Joaquim Reizi Higuchi [view email]<br>[v1]<br>Sat, 15 Mar 2025 14:48:51 UTC (20 KB)

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