category of torsion-free abelian groups

This category is a typical example of a well-behaved additive category which is not abelian. It contains the category of free abelian groups.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: trivial group
  • initial object: trivial group
  • products: direct products
  • coproducts: direct sums

Special morphisms

  • isomorphisms: bijective group homomorphisms
  • monomorphisms: injective group homomorphisms
  • epimorphisms: homomorphisms f:A→Bf : A \to B such that B/f(A)B/f(A) is a torsion group
  • regular monomorphisms: For a homomorphism f:A→Bf : A \to B the following are equivalent: (1) ff is a regular monomorphism. (2) ff is injective and B/f(A)B/f(A) is torsion-free, i.e., ff is the inclusion of a saturated subgroup. (3) ff is injective and whenever we have n∈Z∖{0}n \in \IZ \setminus \{0\} and a∈Aa \in A such that f(a)f(a) is nn-divisible in BB, then aa is nn-divisible in AA. (4) ff is injective and whenever we have pp prime and a∈Aa\in A such that f(a)f(a) is pp-divisible in BB, then aa is pp-divisible in AA.
  • regular epimorphisms: surjective group homomorphisms

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of torsion-free abelian groups. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

Functors

The database stores 1 functor based on the category of torsion-free abelian groups.

Morphisms

The database stores 1 morphism based on the category of torsion-free abelian groups.