# connection on principal bundles over algebraic/geometric stacks

Is there a notion of connection on a principal bundle over an algebraic or geometric stack?

By a geometric stack, I mean a stack over category of manifolds that is representable by a Lie groupoid; that is of the form $$B\mathcal{G}$$ for some Lie groupoid $$\mathcal{G}$$.

As far as I know, the paper (https://arxiv.org/abs/math/0401420) discuss the notion of principal bundles over Lie groupoid and connection on principal bundle over Lie groupoid but does not mention if this notion can be turned into a notion of connection on a principal bundle over a geometric stack.

Are there other versions of connection on a principal bundle over stack?

• see section 4.4 here ://arxiv.org/pdf/0806.1357.pdf – Tsemo Aristide Jan 17 at 15:07
• On first reading, it looks slightly different from what the setup I mentioned above. But, I hope it would be of some use. I will read carefully and respond. – Praphulla Koushik Jan 18 at 2:47
• I fail to see how your notion of principal fibered category is related to the notion of principal bundle over lie groupoid mentioned above... Can you please explain something along those lines – Praphulla Koushik Jan 18 at 17:15
• I saw it late... In page 18, after corollary 3.15, it does say that "Therefore, this allows us to speak about connections and flat connections of a principal bundle over a differentiable stack". But, I still want to look for other references that discuss noton of connection on principal bundle over differentiablr stack.. For that matter, they do not say explicitly what does it mean by a principal bundle over differentiable stack $B\mathcal{G}$. From cor 2.12 I am assuming that, by principal bundle over the stack $B\mathcal{G}$, they mean principal bundle over the Lie groupoid $\mathcal{G}$. – Praphulla Koushik Jan 18 at 18:32