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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 compact objects of the derived category $\text{QCoh}(X)$ are precisely the coherent complexes--that is, those complexes which have bounded and coherent cohomology. Note we need to go to the non-bounded level to speak about compact objects, since our category must have all filtered colimits.

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 compact objects of the derived category $\text{QCoh}(X)$ are precisely the coherent complexes--that is, bounded complexes which have only coherent cohomology. Note we need to go to the non-bounded level to speak about compact objects, since our category must have all filtered colimits.

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).

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 compact objects of the derived category $\text{QCoh}(X)$ are precisely the coherent complexes--that is, those complexes which have bounded and coherent cohomology. Note we need to go to the non-bounded level to speak about compact objects, since our category must have all filtered colimits.

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

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).