Category of abelian groups

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Definition

The category of Abelian groups, sometimes denoted $\operatorname {Ab}$, is defined as follows:

• Its objects are Abelian groups.
• Its morphisms are homomorphisms of groups.
• The identity morphism is defined as the identity map.
• The composition of morphisms is defined by function composition.

Relation with other categories

Functors from this category

• Category of groups: The category of Abelian groups embeds as a full subcategory of the category of groups.
• Category of monoids
• Category of pointed sets
• Category of sets

Functors to this category

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Additional structure

Monoidal structure

• Monoidal category of Abelian groups: This is a monoidal category where the monoidal operation is the tensor product of Abelian groups. Note that the tensor product is neither a product nor a coproduct in the category of Abelian groups.