Editing Natural numbers object (section)

== Equivalent definitions ==

NNOs in [[Cartesian closed category|cartesian closed categories]] (CCCs) or [[Topos|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.{{sfn|Johnstone|2002|loc=A2.5.2}}

<div style="margin-left: 2em">[[Image:NNO definition alt.png|alternate NNO definition]]</div>

This same construction defines weak NNOs in cartesian categories that are not cartesian closed.

In a category with a terminal object 1 and binary [[coproduct]]s (denoted by +), an NNO can be defined as the [[initial algebra]] of the [[endofunctor]] that acts on objects by {{math|{{var|X}} ↦ 1 + {{var|X}}}} and on arrows by {{math|{{var|f}} ↦ id<sub>1</sub> + {{var|f}}}}.<ref>{{Cite book|title=Category theory for computing science|last1=Barr|first1=Michael|last2=Wells|first2=Charles|date=1990|publisher=Prentice Hall|isbn=0131204866|location=New York|oclc=19126000|page=358}}</ref>