nLab
integers object

Integers object

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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) EE with terminal object 11 is

  • there is a unique morphism u:Au : \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 𝒮\mathcal{S} is equivalent to the existence of free groups in 𝒮\mathcal{S}:

Proposition

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

Construction from natural numbers objects

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

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

whereby n:1-n:1\rightarrow\mathbb{Z} is represented by the morphism z:1z:1\rightarrow\mathbb{N} in the n thn^{th} copy of \mathbb{N} appearing in this diagram (starting the count at the 0 th0^{th} copy). The resulting induced map to the colimit

× m:(m,n)nm\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:p : \mathbb{Z} \to \mathbb{Z} (predecessor), defined as p=nsnp = n \circ s \circ n, is an inverse morphism of ss, satisfying the commutative diagram:

It follows that ss and pp 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 0:10:1\rightarrow\mathbb{Z} and 1:11:1\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=0z = 0, s=x+1s = x + 1, and n=n = -.

Last revised on December 23, 2020 at 21:55:21. See the history of this page for a list of all contributions to it.