natural numbers object

natural numbers object (plural natural numbers objects)

1. (category theory) An object which has a distinguished global element (which may be called z, for “zero”) and a distinguished endomorphism (which may be called s, for “successor”) such that iterated compositions of s upon z (i.e., ${\displaystyle s^{n}\circ z}$) yields other global elements of the same object which correspond to the natural numbers (${\displaystyle s^{n}\circ z\leftrightarrow n}$). Such object has the universal property that for any other object with a distinguished global element (call it z’) and a distinguished endomorphism (call it s’), there is a unique morphism (call it φ) from the given object to the other object which maps z to z’ (${\displaystyle \phi \circ z=z'}$) and which commutes with s; i.e., ${\displaystyle \phi \circ s=s'\circ \phi }$.