Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

Question

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous homomorphisms from $G$ to $\mathbb {T}=\mathbb R/\mathbb Z$. Why does the definition specifically involve $\mathbb{T}$? What is special about it and why is the class of continuous homomorphisms from some group to $\mathbb{T}$ more informative (if it is) than to some other group?

Answer 1

These are precisely the unitary one-dimensional representations. Classifying all (complex) one-dimensional representations of a (locally compact) group is the obvious first step in classifying irreducible representations (the building blocks of all reps, in the semisimple situation).

There is another question on thise site asking why we focus on unitary representations, the answer likely being in its convenience: it allows us to generalize the averaging trick for finite groups and to construct the analogue of the group algebra.