Natural numbers object

English

Noun

1. Template:Lb 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., $s^n\circ z$) yields other global elements of the same object which correspond to the natural numbers ($s^n\circ z\leftrightarrow n$). 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’ ($\varphi \circ z=z^{\prime }$) and which commutes with s; i.e., $\varphi \circ s=s^{\prime }\circ \varphi$.