category of abelian groups

 algebra
Notation Ab\Ab Objects abelian groups Morphisms group homomorphisms Parent R−ModR{-}\Mod Related Abfg\Ab_{\fg}, CMon\CMon, FinAb\FinAb, FreeAb\FreeAb, Grp\Grp, TorsAb\TorsAb, TorsFreeAb\TorsFreeAb, grAb\grAb, [(N,≤),Ab][(\IN,\leq),\Ab], AbI\Ab^I, [On,Ab][\On,\Ab] External nLab Link

This category is the prototype of an abelian category. It is the special case of R−ModR{-}\Mod where R=ZR = \IZ.

Satisfied Properties

Assigned properties

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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 with pointwise operations
  • coproducts: direct sums

Special morphisms

  • isomorphisms: bijective morphisms
  • monomorphisms: injective morphisms
  • epimorphisms: surjective morphisms
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: surjective morphisms

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of 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 9 functors based on the category of abelian groups.

Morphisms

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

Symmetric monoidal categories

The database stores 1 symmetric monoidal category based on the category of abelian groups.