Suppose $G$ is a finitely generated group, and suppose $Rep_k(G)$ is its category of representations over some field (or maybe even a ring) $k$, endowed with whatever extra structure is needed --- monoidal structure, fiber functor etc. Is it possible to reconstruct $G$ from this category?

Please note that my question isn't about Tannakian reconstruction. Indeed, Tannakian reconstruction (in the forms familiar to me) requires the tensor category to be rigid (that is, it considers only finite dimensional representations) and produces not $G$ itself, but some proalgebraic group over $k$, called $k$-proalgebraic completion. Proalgebraic completion of a finitely generated group could be trivial: for example, Tarski monster doesn't have any finite dimensional representations.

Thanks in advance!

grouplikeelements: $\{x\;|\;\Delta x=x\otimes x\}$. Over a field, the nonzero grouplike elements recover the group. Invertible grouplike elts for general nonzero ring? Alternately, Tannaka's reconstruction theorem identifies the group with the group of natural automorphisms of the fiber functor. Krein's hypotheses about dualizability are only necessary to identify a category of representations, but you assume that you have one. $\endgroup$ – Ben Wieland Mar 1 '19 at 17:475more comments