natural numbers object

Noun

 * 1)  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 &phi;) from the given object to the other object which maps z to z’ ($$\phi \circ z = z'$$) and which commutes with s; i.e., $$\phi \circ s = s' \circ \phi$$.