I am learning about the classifying space $BG$ of a topological group $G$. I know the definition $$BG=EG/G,$$ where $EG$ is any contractible space on which $G$ acts freely. If I am not mistaken, with this definition we are viewing $BG$ as an object in the homotopy category.
On Wikipedia there is a reference to the functor $F_G$ from the homotopy category to the category os sets, sending a space $Z$ to the set of principal $G$-bundles over $Z$, up to isomorphism.
Question 1. Does the couple $(BG,EG\to BG)$ always represent this functor?
Let us now leave the topological category and move to the category of (group) schemes. I am thinking about the relationship existing in general between $$BG \,\,\,\,\,\,\textrm{and}\,\,\,\,\,\,[\bullet/G].$$ I wrote down what $BG$ is, I wrote down what $[\bullet/G]$ is (I know the definition of it as a stack), and I suspect they should be the same thing. But in the stack world one does usually not divide by automorphisms, so if $BG$ happens to represent the functor $F_G$, then I should conclude (?) that $[\bullet/G]$ is always a scheme.
Question 2. Do we always have $BG=[\bullet/G]$?
(In my questions, "always" means respectively: "for every topological group", and "for every group scheme".) I hope that you can see where my confusion comes from, and that someone will kindly help me to fix it.
Thank you!
In algebraic geometry, if $G$ is a smooth group scheme over $S$, then $BG$ denotes the algebraic stack $[S/G]$ over $S$ (as Alex Youcis remarks in a comment).
By definition, this classifies the "functor in groupoids" which to each $S$-scheme $T$ attaches the groupoid of locally trivial (in either the flat, smooth, or etale topology --- it should be equivalent in this context) principal $G$-bundles over $T$.
If you adopt a simplicial point of view, in which you use simplicial constructions to define stacks and higher stacks, than you can describe $BG$ via the usual simplicial construction.