1.NounAn 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 }
.