Isbell conjugacy and the reflexive completion

TL;DR

This paper explores the reflexive completion of categories, revisiting Isbell conjugacy with new examples, establishing functoriality, and analyzing limits, colimits, and relationships with other completions in a modern categorical framework.

Contribution

It provides a modern categorical analysis of reflexive completion, including new examples, functorial properties, and its relation to Cauchy completion and functor categories.

Findings

Reflexive completion is functorial under certain conditions.

Conditions for categories to have equivalent reflexive completions are identified.

The limits and colimits in reflexive completions are characterized.

Abstract

The reflexive completion of a category consists of the Set-valued functors on it that are canonically isomorphic to their double conjugate. After reviewing both this construction and Isbell conjugacy itself, we give new examples and revisit Isbell's main results from 1960 in a modern categorical context. We establish the sense in which reflexive completion is functorial, and find conditions under which two categories have equivalent reflexive completions. We describe the relationship between the reflexive and Cauchy completions, determine exactly which limits and colimits exist in an arbitrary reflexive completion, and make precise the sense in which the reflexive completion of a category is the intersection of the categories of covariant and contravariant functors on it.