Constellations and their relationship with categories

TL;DR

This paper explores the concept of constellations as a generalization of categories, characterizing their structures, relationships, and how they can be used to construct and analyze various categories and substructures.

Contribution

It provides a comprehensive analysis of constellations, their relationship with categories, and introduces a canonical extension method linking constellations to categories.

Findings

Constellations generalize categories and include many examples.

Categories are special cases of constellations, specifically two-sided constellations.

Abstract

Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories, showing the latter to be special cases of the former. In particular, we characterise those constellations that arise as (sub-)reducts of categories, and show that categories are nothing but two-sided constellations. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation $P$ gives rise to a category $\mathcal{C}(P)$, its canonical extension, in a simplest possible way, and that $P$ is a quotient of $\mathcal{C}(P)$ in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences $\delta$ on a given category $K$ (those for which $K\cong \mathcal{C}((K/\delta)$), and show that the category of constellations is equivalent to the category of categories equipped with distinguished canonical congruence.