Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows \*)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles.


Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)$ and stack $\underline{M}$ whose objects are smooth maps to $M$.


Given Lie group $G$, we have two Lie groupoids associated to it :


* $(G\rightrightarrows \*)$ if we consider Lie group structure.
* $(G\rightrightarrows G)$ if we ignore group structure and treat it as a manifold.


We have corresponding stacks associated :


* $(G\rightrightarrows \*)$ gives stack $B(G\rightrightarrows \*)$, usually denoted by $BG$.
* $(G\rightrightarrows G)$ gives stack $B(G\rightrightarrows G)$, usually denoted by $\underline{G}$.


As any Lie group is a manifold, shouldn't there be some relation with notions $BG$ and $\underline{G}$? I see they are not same. 
How are they related?


It is not even the case that the Lie groupoid $(G\rightrightarrows \*)$ is pull back of $(G\rightrightarrows G)$ or the otherway around.




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I do not know counter part in Algebraic geometry. 


Feel free to (I request you to) relate this to algebraic geometry version of stacks. 



Stack associated to Lie group and manifold
$\underline{G}$ is the homotopy loop space of $BG$.


More precisely, the two terminal maps $G\rightarrow pt$ and $G\rightarrow pt$
yield a weak equivalence $\underline{G} \rightarrow pt\times\_{BG} pt$,
where the right side denotes the homotopy pullback of the diagram $pt\rightarrow BG\leftarrow pt$
and $pt$ denotes the representable stack of a smooth manifold given by a single point.


