Stack associated to Lie group and manifold

Question

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.

Answer 1

$\underline{G}$ 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.