Define the classifying topos of a topological category $C$ as the category of $C$-sheaves on $Obj(C)$. The classifying topos of a discrete group (represented as discrete one-object category) is the category of right $G$-sets.
Is it true that if $G$ is a topological group, represented as a one-object category whose space of morphisms inherits the topology of $G$, then $\mathcal BG$ is the category of topological spaces with continuous right $G$-action?
In my opinion this actually follows from the definition (see e.g. Moerdijk, Classifying spaces and classifying topoi), but I would like to have a confirm since I am just beginning the study of the field.
Thank you in advance.
According to Moerdijk definition, given a category internal $\mathbb C$ to $\mathbf{Top}$ (Moerdijk calls it a topological category but this terminology is quite overloaded already in the litterature), a $\mathbb C$-sheaf is an étale map $$p : S \to \mathbb C_0$$ above the space of object together with a continuous right action $$ S\times_{p,t}\mathbb C_1 \to S $$
In the case of a topological group $G$ seen as a one-object internal category, we got $\mathbb C_0$ isomorphic to a point $1$ and $\mathbb C_1\simeq G$. A $\mathbb C$-sheaf is then a étale map $p : S \to 1$ together with a continous right action $S\times G \to S$. But being étale over a point is being discrete. So $\mathcal B\mathbb C$ is the category of sets $S$ with a right action of $G$ $$ S\times G \to G$$ which is continuous when $S$ is given the discrete topology. This is very far from the category of all topological right actions of $G$. In particular if the group is connected, then $\mathcal B\mathbb C$ is just the category of sets. (This is more or less example (d) below the definition.)
Edit. By the way, this topos is discussed in detail in III.9 of Sheaves in Geometry and Logic.