# Classifying topos of a topological group

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.

• I'm confused. Do you mean the category of sheaves of spaces, rather than $C$-sheaves? The latter doesn't seem to be what you're using in the discrete group example. Feb 27, 2018 at 23:12
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.)