Let $G$ be a finite group. In topological category, we have the quotient stack of a point by $G$, denoted by $[pt/G]$. We also have the classifying space $BG$, which is a topological space.

I am a bit confused as these two notions seems to enjoy the same universal properties.

(1) Viewing them in the homotopy category of topological stacks, are the equivalent?

(2) Are they identical "in the language of infinite categories", if they are, what is the rigorous statement of the fact?

(3) Are there any essential difference between the two? (For example, the moduli stack $\mathcal{M}_g$ and the coarse moduli space $M_g$ are essentially different, as they have different properties, say $\pi_1(M_g)=0$ but $\pi_1(\mathcal{M}_g)$ is the mapping class group.)

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    $\begingroup$ $BG$ has the universal property in the category of spaces, not of stacks. In fact, the universal bundle $pt \to [pt/G]$ is not a pullback of the bundle $EG\to BG$. I don't know enough to say much more though $\endgroup$ Oct 29, 2019 at 15:40


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