I have just started studying stacks. Trying to understand the theory I was thinking about a (very interesting) toy example: $ BG $ the classifying stack of a smooth (over a base scheme $ S $) group G. It is well known that a (smooth) cover is given by its canonical point $ s_0: S \rightarrow BG $ (the trivial torsor over $ S $ seen through Yoneda). Is it true that a map from a scheme $ X $ to $ BG $ factor through $ s_0 $?
In particular I had in mind the concrete example where $ S=\mathrm{Spec}(\mathbb C) $ and where $ G=GL_n $ (so it is true that every torsor is locally trivial in the étale topology).
No, a map $X \to BG$ is determined by a map $f:X \to S$ and a $f^*G$-torsor over $X$, while those map that factor through $s_0$ correspond to trivial torsors. So, as soon as there are nontrivial torsors over $X$, there are maps that do not factor through $s_0$.