A logical analysis of fixpoint theorems

TL;DR

This paper establishes a general fixpoint theorem for contractions in Cauchy-complete quantale-enriched categories, unifying and extending known results across metric, fuzzy, and probabilistic spaces.

Contribution

It introduces a new fixpoint theorem applicable to a broad class of categories with continuous quantales, including conditions for uniqueness.

Findings

Validates the theorem with examples from metric, fuzzy, and probabilistic spaces.

Provides sufficient conditions for fixpoint uniqueness.

Extends classical fixpoint results to enriched categorical contexts.

Abstract

We prove a fixpoint theorem for contractions on Cauchy-complete quantale-enriched categories. It holds for any quantale whose underlying lattice is continuous, and applies to contractions whose control function is sequentially lower-semicontinuous. Sufficient conditions for the uniqueness of the fixpoint are established. Examples include known and new fixpoint theorems for metric spaces, fuzzy metric spaces, and probabilistic metric spaces.