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: 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: 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: 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.
NNOs in cartesian closed categories (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.