1. 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 $\text{Rep}(G)$ 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 $\text{Rep}(G)$.

2. When $G$ is not algebraic, there is still a notion of the classifying space $BG$, the equivalence of categories 

3. hese statements hold at both the abelian and derived level (for the unbounded derived category, one must also impose the condition that