Core Concepts
Cauchy-completeness and the rule of unique choice are closely related in relational doctrines.
Abstract
The article discusses the extension of Cauchy-completeness to enriched categories, focusing on the rule of unique choice. It explores the analogy between Cauchy-completeness and the rule of unique choice in relational doctrines. The paper introduces relational doctrines and their construction with the rule of unique choice. It delves into the concept of Cauchy-complete objects and their significance in relational doctrines. The main results highlight the existence of reflective subcategories based on Cauchy-complete objects. Various examples, including metric spaces, Banach spaces, and compact Hausdorff spaces, are presented to illustrate Cauchy-completeness in different contexts.
Stats
Lawvere vastly generalized the notion of complete metric space to enriched categories.
Cauchy-completeness resembles a formulation of the rule of unique choice.
A relational doctrine defines Cauchy-complete objects as those satisfying the rule of unique choice.
The main result is the existence of a reflector for Cauchy-complete objects in relational doctrines.
Quotes
"Cauchy-completeness resembles a formulation of the rule of unique choice."