nLab integers object

Integers object


Topos Theory

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Integers object


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.


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}:


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}.



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 May 14, 2022 at 00:32:23. See the history of this page for a list of all contributions to it.