The natural numbers in a topos are represented by what is called a natural-numbers object, an object ℕ in the category equipped with a morphism z: 1 → ℕ that serves to pick out the number zero, where 1 is a terminal object in the category, and another morphism s: ℕ → ℕ that satisfies a certain universal free-action property, which ensures that it acts like the successor function on the natural numbers.
