On The Existence Of Category Bicompletions

TL;DR

This paper discusses a conjecture about the conditions under which a category's free small-colimit completion exists and explores implications for the existence of an Isbell-Lambek bicompletion, potentially simplifying its construction.

Contribution

It proposes a conjecture linking the existence of a small generating-cogenerating set to the free small-colimit completion and implications for bicompletion existence.

Findings

Conjecture relates small generating-cogenerating sets to category completions.

Sketches how the conjecture implies the existence of an Isbell-Lambek bicompletion.

Potential to avoid change-of-universe procedures in bicompletion discussions.

Abstract

A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the validity of the result would lead to the existence of an Isbell-Lambek bicompletion C(A) of such an A, without a "change-of-universe" procedure being necessary to describe or discuss the bicompletion.