Poset-enriched categories and free exact completions

TL;DR

This paper presents an elementary method for constructing exact completions of categories enriched in Pos, characterizes these categories via projective objects, and applies the results to ordered algebra varieties, answering a known question.

Contribution

It introduces a new elementary construction for exact completions in Pos-enriched categories and characterizes these categories through projective objects, with applications to ordered algebra varieties.

Findings

Provided an elementary construction for Pos-enriched exact completions.

Characterized categories arising as such completions via projective objects.

Showed that every ordered algebra variety is an exact completion of free algebras.

Abstract

We give an elementary construction of the exact completion of a weakly lex category for categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets. Paralleling the ordinary case, we characterize categories which arise as such completions in terms of projective objects. We then apply the results to categories of Eilenberg-Moore algebras for monads on $\mathsf{Pos}$. In particular, we show that every variety of ordered algebras is the exact completion of a subcategory on certain free algebras, thereby answering a question of A. Kurz and J. Velebil.