Constellations with range and IS-categories

TL;DR

This paper explores constellations with range, showing they are equivalent to ordered categories with restrictions, and introduces IS-categories as their canonical extensions, linking various well-known categories.

Contribution

It characterizes constellations with range as ordered categories with restrictions and introduces IS-categories as their canonical extensions, unifying several category concepts.

Findings

Constellations with range are equivalent to ordered categories with restrictions.

IS-categories contain subcategories of insertions and surjections with unique factorizations.

Category of IS-categories is equivalent to the category of constellations with range.

Abstract

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups. We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.