Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell

TL;DR

This paper characterizes tight limits and colimits in categories, introduces tight completions via nucleus constructions, and shows that these completions preserve existing tight limits and colimits, ensuring idempotency.

Contribution

It introduces the concept of tight limits and colimits, characterizes them, and develops tight completions using nucleus constructions, extending Dedekind-MacNeille ideas to categories.

Findings

Tight limits and colimits can be characterized and approximated.

Tight completions preserve existing tight limits and colimits.

Tight completions are idempotent, generalizing Dedekind-MacNeille completion.

Abstract

While any infimum in a poset can also be computed as a supremum, and vice versa, categorical limits and colimits do not always approximate each other. If I approach a point from below, and you approach it from above, then we will surely meet if we live in a poset, but we may miss each other in a category. Can we characterize the limits and the colimits that approximate each other, and guarantee that we will meet? Such limits and colimits are called *tight*. Some critically important network applications depend on them. This paper characterizes tight limits and colimits, and describes tight completions, derived by applying the nucleus construction to adjunctions between loose completions. Just as the Dedekind-MacNeille completion of a poset preserves any existing infima and suprema, the tight completion of a category preserves any existing tight limits and colimits and is therefore idempotent.