nLab extended natural numbers object

Idea

Recall that a topos is a category that behaves likes the category Set of sets.

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

An extended natural numbers object in any topos (or any cartesian closed distributive category) $E$ with terminal object $1$ and coproduct $A + B$ is

• an object $\overline{\mathbb{N}}$ in $E$

• 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:

$$\array{& A & \overset{q}\rightarrow & 1 + A & \\ \mu_A & \downarrow &&\downarrow & \mu_A\\ &\overline{\mathbb{N}} & \underset{p}\rightarrow& 1 + \overline{\mathbb{N}} & \\ }$$

Thus, the extended NNO is the terminal coalgebra for the endofunctor $A \mapsto 1 + A$ on $E$.

In a symmetric closed distributive monoidal category

The above definition makes sense in any symmetric closed distributive monoidal category $C$, provided that we use the tensor unit and tensor product instead of the terminal object and cartesian product.

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:

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

Examples

• The set of extended natural numbers is the extended natural numbers object in Set

• Given a field $K$, the underlying vector space of the sequence algebra $K^\mathbb{N}$ is the extended natural numbers object in $\mathrm{Vect}_K$.

Properties

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 $\mathbb{N} + 1$.

Related concepts

• extended natural numbers
• natural numbers object