I am reading "[Some notes on Differentiable stacks](https://www.uni-due.de/%7Ehm0002/stacks.pdf)" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.



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> Let $G$ be a Lie group action on a manifold $X$ (left action). We define the quotient stack $[X/G]$ as $[X/G](Y):=\{P\xrightarrow{p} Y, P\xrightarrow{f}X | P\rightarrow Y \text{ is a G-bundle,} ~ f \text{ is } G\text{-equivariant}\}$.
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> Morphisms of objects are $G$-equivariant isomorphisms.
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I am trying to understand the motivation for defining in this way.




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Given a Lie group action of $G$ on $X$, if I want to associate a stack, I would start with simpler cases which allows me to guess how to define.


* Given a Lie group $G$, I have stack associated to it, denoted by $BG$, the stack of principal $G$ bundles.
* Given a manifold $M$, I have the stack associated to it, denoted by $\underline{M}$ whose objects are maps $Y\rightarrow M$.


Suppose $X$ is trivial and $G$ acts trivially on $X=\{\*\}$ then $[X/G]$ should only depend on $G$. We know what stack to associate for a Lie group $G$ i.e., $BG$. Thus, $[X/G]$ should just be $BG$.


Suppose $G$ is trivial and $G$ acts on $X$, $[X/G]$ should only depend on $X$. We know what stack to associate for a manifold $X$ i.e., $\underline{X}$. Thus, $[X/G]$ should just be $\underline{X}$.


Suppose $G$ is non trivial and $X$ is non trivial and that the action of $G$ on $X$ is such that $X/G$ is a manifold. We know what stack to associate for a manifold $X/G$ i.e., $\underline{X/G}$. Thus, $[X/G]$ should just be $\underline{X/G}$.


I am not able to guess how could we guess the definition knowing above three cases. Is quotient stack definition motivated from these simpler cases or Is it the case that simpler cases are special cases of notion of Quotient stack. How could we come up with such a definition. Any comments regarding the motivation are welcome.



Motivation for definition of Quotient stack
Let's start approaching the question from the simplest possible case $Y=\*$. What should be the points of $[X/G]$?


Recall that the idea here is to generalize the construction of the [*action groupoid*](https://ncatlab.org/nlab/show/action+groupoid) for discrete groups acting on sets to the manifold case. This allows us to remember the stabilizers of points and it is generally a much better behaved notion.


So morally $[X/G](\*)$ should be the groupoid whose objects are points of $X$ and such that $\mathrm{Mor}(x,x')=\{g\in G\mid gx=x'\}$. With this description however it is a bit unclear how to generalize that to get a description of $[X/G](Y)$, so let us rewrite it in a slightly different way.


A point of $X$ is just a $G$-equivariant morphism $G\to X$ (since any such $G$-equivariant morphism is determined by the image of $e\in G$). Moreover a morphism between $x:G\to X$ and $x':G\to X$ is exactly a $G$-equivariant morphism $g:G\to G$ (i.e. right multiplication by some element $g\in G$) making the obvious diagram commute. Now if you look at the definition, the groupoid does not depend from the fact that $G$ has a canonical basepoint (the identity element $e\in G$), so in fact we can write



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> $[X/G](\*)$ is defined as the groupoid of $G$-equivariant maps $T\to X$ where $T$ is a freely transitive $G$-space.
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Ok, so now we want to describe the groupoid $[X/G](Y)$. Intuitively the objects here should be families of elements of $[X/G](\*)$ parametrized by $Y$. But a family of freely transitive $G$-spaces is exactly a principal $G$-bundle $P\to Y$, and a family of $G$-equivariant map $P\_y\to X$ for each $y\in Y$ is just a $G$-equivariant map $P\to X$. Hence we get the definition you are asking about.


