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In category theory, a natural numbers obje … 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: 1.
* a global element z : 1 → N, and 2.
* 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: 1.
* u ∘ z = q, and 2.
* u ∘ s = f ∘ u. 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: 1.
* ⊢ u (z) = q 2.
* y ∈E N ⊢ u (s y) = 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.not required, then N is called a weak NNO.
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| rdfs:comment |
In category theory, a natural numbers obje … 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: 1.
* a global element z : 1 → N, and 2.
* 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: 1.
* u ∘ z = q, and 2.
* u ∘ s = f ∘ u. In other words, the triangle and square in the following diagram commute. 1.
* ⊢ u (z) = q 2.
* y ∈E N ⊢ u (s y) = f (u (y))(z) = q 2.
* y ∈E N ⊢ u (s y) = f (u (y))
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