最終更新日：2025/12/05
(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.
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natural numbers object
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元となった辞書の項目
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
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.
（ plural ）
