# Stack associated to Lie group and manifold

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.

I do not know counter part in Algebraic geometry.

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

• Can some one up voting the question leave a message :P :D – Praphulla Koushik Feb 20 '19 at 18:03
• I don't understand what is the utility of "Are they same? I see they are not same." – LSpice Feb 20 '19 at 18:13
• @LSpice I have edited it. :) Bad English skills :) – Praphulla Koushik Feb 20 '19 at 18:17
• It's nothing to do with algebraic geometry, just pure stack theory. – David Roberts Feb 20 '19 at 21:29
• @DavidRoberts I though similar question can be asked from algebraic geometry perspective so that it will be convenient for them to think.. – Praphulla Koushik Feb 21 '19 at 4:52

$$\underline{G}$$ 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.
• I will say what I understand.. For Lie group $G$, we have obvious map of stacks $\underline{*}\rightarrow BG$ (which is actually an atlas for $BG$). Consider the $2$-fibre product $\underline{*}\times_{BG}\underline{*}$. Here, $*$ denote singleton space and $\underline{*}$ denote the stack associated to singleton manfold. What does it mean to say a weak equivalence $G\rightarrow \underline{*}\times_{BG}\underline{*}$.. I only know weak (Morita ??) equivalence in case of Lie groupoids.. Does it mean weak equivalence as in page 9 of maths.qmul.ac.uk/~noohi/papers/quick.pdf? – Praphulla Koushik Feb 20 '19 at 19:34
• searching in google said that, "loop space" of space $BG$ is a space that is homotopy equivalent to $G$ (math.stackexchange.com/questions/442805/…) Can you please tell me How should I relate with my question? – Praphulla Koushik Feb 20 '19 at 19:47