In a category $\mathcal{A}$ (with all filtered colimits, like $\text{Rep}(G)$), one can define the notion of a *compact object* $A \in \mathcal{A}$ as an object for which the functor $\text{Hom}_{\mathcal{A}}(A, -)$ commutes with all filtered colimits. Since objects of the abelian category $\text{Rep}(G)$ are direct sums of their irreducible components, it's easy to verify directly that an object is compact if and only if it is finite dimensional. Therefore, the finite dimensional representations are precisely the compact objects of $\text{Rep}(G)$. 

Similar assertions to the above hold at the derived category level. For a regular scheme $X$, the bounded complexes on $X$ with coherent cohomology are precisely the compact objects of $\mathcal{D}^{b}\text{QCoh}(X)$ (for the unbounded derived category, one must also throw in that the cohomologies eventually vanish).