Partial orders are the free conservative cocompletion of total orders

TL;DR

This paper establishes that the category of posets can be constructed as the free conservative cocompletion of the simplex category, linking partial orders to total orders through a universal categorical process.

Contribution

It demonstrates that the category of posets is the free conservative cocompletion of the simplex category, providing a new categorical perspective on partial and total orders.

Findings

Posets are equivalent to the free conservative cocompletion of the simplex category.

The result connects partial orders with total orders via a universal categorical construction.

This provides a foundational categorical characterization of posets in relation to total orders.

Abstract

We show that the category of partially ordered sets $\mathsf{Pos}$ is equivalent to the free conservative cocompletion of the category of finite non-empty totally ordered sets $\Delta$, which is also known as the simplex category.