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 of integers does in Set.
An integers object in a topos (or any cartesian closed category) with terminal object is
By the universal property, the integers object is unique up to isomorphism.
The existence of an integers object in a topos is equivalent to the existence of free groups in :
Let be a topos and its category of internal group objects. Then has an integers object precisely if the forgetful functor has a left adjoint.
Suppose the topos has a natural numbers object . Then an integers object is a filtered colimit of objects
whereby is represented by the morphism in the copy of appearing in this diagram (starting the count at the copy). The resulting induced map to the colimit
imparts a monoid structure (in fact a group structure) on descended from the monoid structure on .
The morphism (predecessor), defined as , is an inverse morphism of , satisfying the commutative diagram:
It follows that and are both isomorphisms of .
In a category with finite products, the initial ring object, an object with global elements and , a morphism , morphsims and , and suitable commutative diagrams expressing the ring axioms and initiality, has the structure of an integers object given by , , and .
Last revised on May 14, 2022 at 04:32:23. See the history of this page for a list of all contributions to it.