natural numbers object
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(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., sⁿ∘z) yields other global elements of the same object which correspond to the natural numbers (sⁿ∘z↔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') and which commutes with s; i.e., 𝜙∘s=s'∘𝜙.
natural numbers object
In category theory, a natural numbers object is an object equipped with a distinguished global element z and a distinguished endomorphism s such that the iterated compositions sn ∘ z yield the natural numbers, and it satisfies the universal property that for any other object with a distinguished global element z' and endomorphism s', there is a unique morphism φ with φ ∘ z = z' and φ ∘ s = s' ∘ φ.
In category theory, a natural numbers object is an object equipped with a distinguished global element z and a distinguished endomorphism s such that the iterated compositions sn ∘ z yield the natural numbers, and it satisfies the universal property that for any other object with a distinguished global element z' and endomorphism s', there is a unique morphism φ with φ ∘ z = z' and φ ∘ s = s' ∘ φ.
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