Objects are abelian groups and morphisms are group homomorphisms. Addition on hom-sets is pointwise, kernels are subgroup kernels, and cokernels are quotients by images. Thus this is an abelian category. Products are Cartesian products; coproducts are direct sums of finite-support families. The underlying-set functor creates directed colimits, giving a concrete description of their elements by eventual equality of representatives.
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The category of abelian groups, often denoted as \(\mathbf{Ab}\), is a mathematical structure in category theory that consists of abelian groups as objects and group homomorphisms as morphisms. Here's a more detailed breakdown of its features: 1. **Objects**: The objects in \(\mathbf{Ab}\) are all abelian groups.