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 $1$;

  • an isomorphism $s : \mathbb{Z} \cong \mathbb{Z}$ (successor),

such that for all objects $A$ with morphism $z_A:1 \to A$ and isomorphism $s_A:A \cong A$, there is a unique morphism $u_A:\mathbb{Z} \to A$ such that $u_A \circ z = z_A$ and $u_A \circ s = s_A \circ u_A$.

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

In a closed symmetric monoidal category

One could generalize the above definition of an integers object to any closed symmetric monoidal category: pointed objects in a symmetric monoidal category are represented by morphisms out of the tensor unit. Thus, an integers object in a closed symmetric monoidal category $C$ with tensor unit $1$ is

• an object $\mathbb{Z}$ in $C$

• equipped with

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

  • an isomorphism $s : \mathbb{Z} \cong \mathbb{Z}$ (successor);

such that for all objects $A$ with morphism $z_A:1 \to A$ and isomorphism $s_A:A \cong A$, there is a unique morphism $u_A:\mathbb{Z} \to A$ such that $u_A \circ z = z_A$ and $u_A \circ s = s_A \circ u_A$.

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 $\mathbb{N} \times \mathbb{N}$.

Examples

There are many examples of integers objects.

In closed symmetric monoidal categories

Example

The integers are the integers object in the closed symmetric monoidal category Set.

Example

In classical mathematics, the Alexandroff compactification of the integers, $\mathbb{Z}^* = \mathbb{Z} + \{\infty\}$, is the integers object in the closed symmetric monoidal category of pointed sets $Set_*$, with $z:\mathbb{2} \to \mathbb{Z}^*$ taking the boolean true to $\infty$ and false to $0$ and $s:\mathbb{Z}^* \cong \mathbb{Z}^*$ taking integers to its successor and $\infty$ to $\infty$. In constructive mathematics, the Alexandroff compactification of the integers and the disjoint union $\mathbb{Z} + \{\infty\}$ are no longer the same; it is $\mathbb{Z} + \{\infty\}$ which remains the integers object in $Set_*$.

Example

The underlying abelian group of the ring of Laurent polynomials $\mathbb{Z}[X, X^{-1}]$ is the integers object in the closed symmetric monoidal category Ab, with $z:\mathbb{Z} \to \mathbb{Z}[X, X^{-1}]$ taking integers to constant Laurent polynomials and the abelian group isomorphism $s:\mathbb{Z}[X, X^{-1}] \cong \mathbb{Z}[X, X^{-1}]$ multiplying Laurent polynomials by the indeterminant $X$.

Example

More generally, given a commutative ring $R$, the underlying $R$-module of the ring of Laurent polynomials $R[X, X^{-1}]$ is the integers object in the closed symmetric monoidal category RMod, with $z:R \to R[X, X^{-1}]$ taking scalars to constant Laurent polynomials and the linear isomorphism $s:R[X, X^{-1}] \cong R[X, X^{-1}]$ multiplying Laurent polynomials by the indeterminant $X$.

In a sheaf topos

In any Grothendieck topos $E = Sh(C)$ the integers object is given by the constant sheaf on the set of ordinary integers, i.e. by the sheafification of the presheaf $C^{op} \to Set$ that is constant on the set $\mathbb{Z}$.

Similar to the case for natural numbers objects, there are interesting cases in which such sheaf toposes contain objects that look like they ought to be integers objects but do not satisfy the above axioms: for instance some of the models described at Models for Smooth Infinitesimal Analysis are sheaf toposes that contain besides the standard integers object a larger object of smooth integers that has generalized elements which are “infinite integers” in the sense of nonstandard analysis.

Properties

Inverse

By definition of isomorphism, there is an inverse isomorphism $p:\mathbb{Z} \cong \mathbb{Z}$, where $p \circ s = \mathrm{id}_\mathbb{Z}$ and $s \circ p = \mathrm{id}_\mathbb{Z}$.

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 $z = 0$ and $s = x + 1$.

Related concepts

• integers

• integers type

• natural numbers object

• free group

• ring object

• circle object