One categorical definition of a group $G$ is that it is a category $C$ with a single object $X$ such that every morphism in the set $C(X,X)$ is invertible, i.e. such that $C(X,X)$ is precisely the set-theoretical group $G$. Using this definition, a representation of the group $G$ is a functor from $C$ to the category of vector spaces.
Now there is another categorical definition of a group, which is perhaps less strange, and perhaps more illuminating -- namely that a group $G$ is a group object in the category of sets. This definition has the advantage that a group object can be defined relative any category that admits finite products, so in particular it can be defined relative to, say, the category of topological spaces, thus giving us a topological group.
My question then concerns the relationship between the two definitions: namely, if we fix an arbitrary cartesian monoidal category $M$, then I believe that a group object $G$ in $M$ gives rise to an $M$-enriched category $C$ with a single object $X$ with $C(X,X)$ -- the group object $G$ in $M$. Is this correct?
Furthermore, does this mean that we can define some sort of categorical representations of a group object $G$ in a cartesian monoidal category $M$ as the functors from $C$ to some fixed $M$-enriched category? So in particular we can treat representations of topological groups from this perspective (since I think any category enriched over the category of sets is enriched over the category of topological spaces as the former embeds faithfully into the latter)?