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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)?

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    I think the main obstruction to your approach is that the category of topological spaces is not enriched over itself in an natural way: it fails to be cartesian closed. Moreover isn't a representation of a topological group supposed to be a _continuous_ map $G \times V \to V$, where $V$ is a _topological_ vector space?2012-03-19

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