natural numbers object

Noun

• 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., $s^n\; \backslash circ\; z$) yields other global elements of the same object which correspond to the natural numbers ($s^n\; \backslash circ\; z\; \backslash leftrightarrow\; n$). 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’ ($\backslash phi\; \backslash circ\; z\; =\; z\text{'}$) and which commutes with s; i.e., $\backslash phi\; \backslash circ\; s\; =\; s\text{'}\; \backslash circ\; \backslash phi$.