A topological abelian group (or TAG) is a topological group whose underlying group is abelian.
Topological abelian groups and continuous group homomorphisms form a category $Ab Top$.
Often TAGs are easier to understand than arbitrary topological groups, because no distinction need be made between left and right. For example, a TAG has only one uniform structure, a locally compact TAG has only one Haar measure (up to scale), etc.
TAGs are important in analysis. For example, every topological vector space (TVS) is a TAG, and much of the theory of TVSs can be generalised to TAGs. See also G-norm.
Jiri Adamek, Horst Herrlich, and George Strecker, Abstract and concrete categories: the joy of cats. free online
HAF (for applications to analysis)