In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO N is given by:
a global element z : 1 → N, and
an arrow s : N → N,
such that for any object A of E, global element q : 1 → A, and arrow f : A → A, there exists a unique arrow u : N → A such that:
u ∘ z = q, and
u ∘ s = f ∘ u.[1][2][3]
In other words, the triangle and square in the following diagram commute.
The pair (q, f) is sometimes called the recursion data for u, given in the form of a recursive definition:
⊢ u (z) = q
y ∈EN ⊢ u (sy) = f (u (y))
The above definition is the universal property of NNOs, meaning they are defined up to canonical isomorphism. If the arrow u as defined above merely has to exist, that is, uniqueness is not required, then N is called a weak NNO.
^Johnstone 2002, A2.5.1.
^Lawvere 2005, p. 14.
^Leinster, Tom (2014). "Rethinking set theory". American Mathematical Monthly. 121 (5): 403–415. arXiv:1212.6543. Bibcode:2012arXiv1212.6543L. doi:10.4169/amer.math.monthly.121.05.403. S2CID 5732995.
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