I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's *Algebraic Stacks and Moduli of Vector Bundles*. 


Explicitly, let $G$ be an affine smooth group $S$-scheme with right action $\rho:X\times G\to X$ on a noetherian $S$-scheme $X$. The **quotient stack** $[X/G]$ is the pseudofunctor
$$
[X/G]:(\mathsf{Sch}/S)^{\operatorname{op}}\to\mathsf{Grpds}
$$
defined as follows. 


For an $S$-scheme $U$ let $[X/G](U)$ be the category whose objects are diagrams
$$
U\xleftarrow{\pi}E\xrightarrow{\alpha}X
$$
where $\pi$ is a principal $G$-bundle and $\alpha$ is a $G$-equivariant morphism. The morphisms in $[X/G](U)$ are the isomorphisms of principal $G$-bundles commuting with the $G$-equivariant morphisms.


For a morphism of $S$-schemes $f:U^\prime\to U$ let
$$
[X/G](f):[X/G](U)\to[X/G](U^\prime)
$$
be the functor induced by pullbacks of principal $G$-bundles.


It is not clear to me how to determine when $[X/G]$ is representable in general. My questions are:


**Question 1.** Is there a sufficient condition we can impose on $\rho$ to ensure that $[X/G]$ is representable? The [Wikipedia article](http://en.wikipedia.org/wiki/Quotient_stack) on quotient stacks says something about when the categorical quotient $X/G$ exists the canonical map $[X/G]\to\operatorname{Hom}(-,X/G)$ need not necessarily be an isomorhpism. This is somewhat opaque to me as I'm not even sure how $[X/G]\to\operatorname{Hom}(-,X/G)$ is defined.


**Question 2.** What is an example where $[X/G]$ is *not* representable?


If these questions are too broad, I'd be very grateful if someone could point me in the direction of a good reference.



Understanding the definition of the quotient stack $[X/G]$
The functor $[X/G]$ is not representable whenever there is non-trivial isotropy of the action of $G$ on $X$.


Let us consider the most extreme case: when $X = \bullet$ is a point (the terminal object) and $G$ is any non-trivial group. In such a case, $Hom(-,\bullet/G)$ is a singleton, as $\bullet / G = \bullet$, which is still terminal.


However, $[\bullet / G]$ is much more interesting than that! If you trace through the definition provided above (which is worth doing at least once in your life), we find that $[\bullet / G](U)$ is the collection of principal $G$-bundles over $U$, which is non-trivial most of the time. Thus it follows that, as a stack, $[\bullet / G]$ is the classifying space of $G$.


Particular examples are $[\bullet / \mathbb{G}\_m]$ being the collection of line bundles, etc.


As for the question about how the morphism $[X/G] \to Hom(-,X/G)$ is defined:


In your diagram, the morphism $\alpha : E \to X$ is equivariant. You can thus complete the diagram to


$$
\begin{array}{c c c}
E & \to & X \\
\downarrow & & \downarrow\\
U & \to & X/G
\end{array}
$$


which yields the desired map.


