nLab integers object

Idea

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

An integers object internal to a topos is an object that behaves in that topos like the set $\mathbb{Z}$ of integers does in Set.

Definition

In a topos or cartesian closed category

An integers object in a topos (or any cartesian closed category) $E$ with terminal object $1$ is

• an object $\mathbb{Z}$ in $E$

• equipped with

  • a morphism $z:1 \to \mathbb{Z}$ from the terminal object $1$;

  • a morphism $s : \mathbb{Z} \to \mathbb{Z}$ (successor),

  • a morphism $n : \mathbb{Z} \to \mathbb{Z}$ (negation)

• such that the following diagram commutes:

• and that for every other commutative diagram of form

• there is a unique morphism $u : \mathbb{Z} \to A$ such that the following diagram commutes

By the universal property, the integers object is unique up to isomorphism.

Free construction in a topos

The existence of an integers object in a topos $\mathcal{S}$ is equivalent to the existence of free groups in $\mathcal{S}$:

Proposition

Let $\mathcal{S}$ be a topos and $\mathbf{Grp}(\mathcal{S})$ its category of internal group objects. Then $\mathcal{S}$ has an integers object precisely if the forgetful functor $U:\mathbf{Grp}(\mathcal{S})\to \mathcal{S}$ has a left adjoint.

Construction from natural numbers objects

Suppose the topos $E$ has a natural numbers object $\mathbb{N}$. Then an integers object $\mathbb{Z}$ is a filtered colimit of objects

$$\mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \ldots$$

whereby $-n:1\rightarrow\mathbb{Z}$ is represented by the morphism $z:1\rightarrow\mathbb{N}$ in the $n^{th}$ copy of $\mathbb{N}$ appearing in this diagram (starting the count at the $0^{th}$ copy). The resulting induced map to the colimit

$$\mathbb{N} \times \mathbb{N} \cong \sum_{m \in \mathbb{N}} \mathbb{N} \to \mathbb{Z}: (m, n) \mapsto n-m$$

imparts a monoid structure (in fact a group structure) on $\mathbb{Z}$ descended from the monoid structure on $\mathbb{N} \times \mathbb{N}$.

Properties

Inverse

The morphism $p : \mathbb{Z} \to \mathbb{Z}$ (predecessor), defined as $p = n \circ s \circ n$, is an inverse morphism of $s$, satisfying the commutative diagram:

It follows that $s$ and $p$ are both isomorphisms of $\mathbb{Z}$.

Initial ring object

In a category with finite products, the initial ring object, an object $\mathbb{Z}$ with global elements $0:1\rightarrow\mathbb{Z}$ and $1:1\rightarrow\mathbb{Z}$, a morphism $-:\mathbb{Z}\rightarrow\mathbb{Z}$, morphsims $+:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$ and $\times:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$, and suitable commutative diagrams expressing the ring axioms and initiality, has the structure of an integers object given by $z = 0$, $s = x + 1$, and $n = -$.

Related concepts

• integers

• natural numbers object

• free group

• ring object

• circle object