In the following everything is over some field $k$.

Let $G$ be a discrete group. We write $G^{\text{alg}}$ for its *pro-algebraic completion*. The latter is a pro-affine pro-algebraic group which arises as the Tannakian dual to the category $\mathrm{Rep}_{\text{fd}}(G)$ of finite-dimensional $G$-representations. By definition there is an equivalence of categories
$$
\mathrm{Rep}_{\text{fd}}(G)\simeq \mathrm{Rep}_{\text{fd}}(G^\text{alg}).
$$
Under what conditions does this lift to an equivalence of categories of possibly-infinite dimensional representations?

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