Editing Natural numbers object (section)

== Examples ==

* In '''Set''', the [[category of sets]], the standard natural numbers are an NNO.{{sfn|Johnstone|2002|p=108}} A terminal object in '''Set''' is a [[Singleton (mathematics)|singleton]], and a function out of a singleton picks out a single [[Element (set theory)|element]] of a set. The natural numbers 𝐍 are an NNO where {{var|z}} is a function from a singleton to 𝐍 whose [[Image (mathematics)|image]] is zero, and {{var|s}} is the [[successor function]]. (We could actually allow {{var|z}} to pick out ''any'' element of 𝐍, and the resulting NNO would be isomorphic to this one.) One can prove that the diagram in the definition commutes using [[mathematical induction]].
* In the category of types of [[Martin-Löf type theory]] (with types as objects and functions as arrows), the standard natural numbers type '''nat''' is an NNO. One can use the recursor for '''nat''' to show that the appropriate diagram commutes.
* Assume that <math> \mathcal{E} </math> is a [[Grothendieck topos]] with terminal object <math> \top </math> and that <math> \mathcal{E} \simeq \mathbf{Shv}(\mathfrak{C},J) </math> for some [[Grothendieck topology]] <math> J </math> on the category <math> \mathfrak{C} </math>. Then if <math> \Gamma_{\mathbb{N}} </math> is the constant presheaf on <math> \mathfrak{C} </math>, then the NNO in <math> \mathcal{E} </math> is the sheafification of <math> \Gamma_{\mathbb{N}} </math> and may be shown to take the form <math display="block"> \mathbb{N}_{\mathcal{E}} \cong \left(\Gamma_{\mathbb{N}}\right)^{++} \cong \coprod_{n \in \mathbb{N}} \top. </math>