natural numbers object

== English == === Noun === natural numbers object (plural natural numbers objects) (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., s n ∘ z {\displaystyle s^{n}\circ z} ) yields other global elements of the same object which correspond to the natural numbers ( s n ∘ z ↔ n {\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’ ( ϕ ∘ z = z ′ {\displaystyle \phi \circ z=z'} ) and which commutes with s; i.e., ϕ ∘ s = s ′ ∘ ϕ {\displaystyle \phi \circ s=s'\circ \phi } .