An extended natural numbers object (extended NNO) in a topos is an object that behaves internal to that topos like the set $\overline{\mathbb{N}}$ of extended natural numbers does in Set.

Definition

In a topos or cartesian closed distributive category

equipped with a morphism $p:\overline{\mathbb{N}} \to (1 + \overline{\mathbb{N}})$

such that for every other object $A$ with morphism $q:A \to (1 + A)$, there is a unique morphism $\mu_A:A \to \overline{\mathbb{N}}$ such that this diagram commutes:

An extended natural numbers object in a symmetric closed distributive monoidal category $C$ with tensor unit$I$ and coproduct$A \oplus B$ is

an object $\overline{N}$ in $C$

equipped with a morphism $p:\overline{N} \to (I \oplus \overline{N})$

such that for every other object $A$ with morphism $q:A \to (I \oplus A)$, there is a unique morphism $\mu_A:A \to \overline{N}$ such that this diagram commutes:

$\array{& A & \overset{q}\rightarrow & I \oplus A & \\
\mu_A & \downarrow &&\downarrow & \mu_A\\
&\overline{N} & \underset{p}\rightarrow& I \oplus \overline{N} & \\
}$

If the topos is a Boolean topos with a natural numbers object, then the extended natural numbers object $\overline{\mathbb{N}}$ is isomorphic to the object $\mathbb{N} + 1$.