Constellations and their relationship with categories

Abstract

Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be 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. In particular, we characterise those constellations that arise as (sub-)reducts of categories. 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 ${\delta}$-categories, that is, categories equipped with distinguished canonical congruence ${\delta}$.

The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are ${\delta}$-categories) may be subsumed by constellation theory.