Let
be a category with a terminal object. A (weak) natural numbers object consists of an object along with morphisms (“zero”) and
(“successor”) of
such that if
is a diagram in
, then there exists a morphism
such that the diagram
is commutative. The morphism is not required to be unique. If we additionally require the morphism to be unique, we obtain a strong natural numbers object. Note that a strong natural numbers object can also be defined as an initial diagram of the form
Example In the category
, the set
of natural numbers is a natural numbers object. Using arithmetic notation for simplicity, the morphism
picks out the element zero, and the morphism
is defined by the formula
. This object is the source of the name “natural numbers object”.
![$ \xymatrix{1\ar[r]^{z'} & N'\ar[r]^{S'} & N}$](http://images.planetmath.org:8080/cache/objects/8806/l2h/img7.png "$ \xymatrix{1\ar[r]^{z'} & N'\ar[r]^{S'} & N}$")
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