  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.

Category Theory