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Internal Group Actions as Enriched Functors

Earlier today this month on the Category Theory Zulip, Bernd Losert asked an extremely natural question about how we might study topological group actions via the functorial approach beloved by category theorists. The usual story is to treat a group $G$ as a one-object category $\mathsf{B}G$. Then an action $G \curvearrowright X$ is the same data as a functor $\mathsf{B}G \to \mathsf{Set}$ sending the unique object of $\mathsf{B}G$ to $X$. Is there some version of this story that works for topological groups and continuous group actions?

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