dual of the category of abelian groups

 algebra
Notation Abop\Ab^{\op} Objects abelian groups Morphisms A morphism A→BA \to B is a group homomorphism B→AB \to A. External nLab link Dual Ab\Ab
By definition, this category is the dual (or opposite) of the category Ab\Ab. By Pontryagin duality, it is equivalent to the category CompHausAb\CompHausAb of compact Hausdorff abelian groups with continuous homomorphisms. The equivalence maps an abelian group AA to its character group Hom⁡(A,R/Z)\Hom(A,\IR/\IZ), equipped with the topology of pointwise convergence, and a compact Hausdorff abelian group KK to the group of continuous homomorphisms K→R/ZK \to \IR/\IZ.

Satisfied properties

Assigned properties

—

Deduced properties

Unsatisfied properties

Assigned properties

—

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

—

Special objects

  • terminal object: trivial group
  • initial object: trivial group
  • products: direct sums
  • coproducts: direct products with pointwise operations

Special morphisms