[[Group theory MOC]]
# Category of abelian groups

The **category of abelian groups** $\Ab$ is a [[Subcategory|full subcategory]] of [[Category of groups]] containing abelian groups and homomorphisms between them.
The category is equivalent to [[Category of left modules|$\lMod{\mathbb{Z}}$]] ([[Abelian groups as Z-modules]]).

## Properties

- There is a [[Free-forgetful adjunction]] between the [[Abelianization]] $(-)^\mathrm{ab} : \Grp \to \Ab$ and the inclusion functor $\mathrm{I} : \Ab \to \Grp$.
- Both the finitary product and coproduct is the [[Direct product of groups]], hence it is an [[Additive category]]

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