English Dictionary

natural numbers object

名詞

(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.

日本語の意味

圏論において、特定の全体元（例えば“0”を示すz）と特定の自己写像（例えば“後続”を示すs）を持つ対象を指す。この対象では、sを繰り返し適用することでzから得られる系列の元が自然数に対応し、かつ任意の同様の構造（全体元z'と自己写像s'）を持つ対象に対して、zをz'に写し、sとs'の関係を保つ一意な射が定義されるという普遍性を満たす

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.

圏論において、自然数対象とは、終対象からの指定された点 z と指定された自己射 s を備え、s^n ∘ z の反復が自然数に対応し、さらに任意の他の対象に指定された点 z' と自己射 s' が与えられるときに z を z' に写し s と可換な一意の射 φ が存在するという普遍性を満たす対象である。

意味（1）

(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 objects （ plural ）