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- Natural_number_object abstract "In category theory, a natural number 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 (alternately, a topos), 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.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 ∈E N ⊢ u (s y) = f (u (y))NNOs are defined up to isomorphism. Every NNO is an initial object of the category of diagrams of the formIf the arrow u as defined above merely has to exist, i.e. uniqueness is not required, then N is called a weak NNO. If a cartesian closed category has weak NNOs, then every slice of it also has a weak NNO. NNOs in CCCs or topoi are sometimes defined in the following equivalent way (due to Lawvere): for every pair of arrows g : A → B and f : B → B, there is a unique h : N × A → B such that the squares in the following diagram commute.This same construction defines weak NNOs in cartesian categories that are not cartesian closed.NNOs can be used for non-standard models of type theory in a way analogous to non-standard models of analysis. Such categories (or topoi) tend to have "infinitely many" non-standard natural numbers. (Like always, there are simple ways to get non-standard NNOs; for example, if z = s z, in which case the category or topos E is trivial.)Freyd showed that z and s form a coproduct diagram for NNOs; also, !N : N → 1 is a coequalizer of s and 1N, i.e., every pair of global elements of N are connected by means of s; furthermore, this pair of facts characterize all NNOs.".
- Natural_number_object thumbnail NNO_definition.png?width=300.
- Natural_number_object wikiPageID "918609".
- Natural_number_object wikiPageRevisionID "520260205".
- Natural_number_object hasPhotoCollection Natural_number_object.
- Natural_number_object subject Category:Objects_(category_theory).
- Natural_number_object type Abstraction100002137.
- Natural_number_object type DefiniteQuantity113576101.
- Natural_number_object type Measure100033615.
- Natural_number_object type NaturalNumber113728367.
- Natural_number_object type NaturalNumbers.
- Natural_number_object type Number113582013.
- Natural_number_object comment "In category theory, a natural number object (NNO) is an object endowed with a recursive structure similar to natural numbers.".
- Natural_number_object label "Natural number object".
- Natural_number_object sameAs m.03pv7k.
- Natural_number_object sameAs Q6980740.
- Natural_number_object sameAs Q6980740.
- Natural_number_object sameAs Natural_number_object.
- Natural_number_object wasDerivedFrom Natural_number_object?oldid=520260205.
- Natural_number_object depiction NNO_definition.png.
- Natural_number_object isPrimaryTopicOf Natural_number_object.
