Return to Question

(rough) Definition : A bibundle is a groupoid principal bundle which is equipped with a second groupoid action from the other side.

 

(precise) Definition : A bibundle from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with maps $a_L:P\rightarrow \mathcal{G}_0$ and $a_R:P\rightarrow \mathcal{H}_0$ such that

 

1. there is a left action of $\mathcal{G}$ on $P$ with respect to an anchor $a_L$ and a right action of $\mathcal{H}$ on $P$ with respect to an anchor $a_R$.

2. $a_L:P\rightarrow \mathcal{G}_0$ is a principal $H$-bundle.

3. $a_R$ is $\mathcal{G}$ invariant.

4. the actions of $\mathcal{G}$ and $\mathcal{H}$ commutes.

(rough) Definition : A bibundle is a groupoid principal bundle which is equipped with a second groupoid action from the other side.

(precise) Definition : A bibundle from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with maps $a_L:P\rightarrow \mathcal{G}_0$ and $a_R:P\rightarrow \mathcal{H}_0$ such that

1. there is a left action of $\mathcal{G}$ on $P$ with respect to an anchor $a_L$ and a right action of $\mathcal{H}$ on $P$ with respect to an anchor $a_R$.

2. $a_L:P\rightarrow \mathcal{G}_0$ is a principal $H$-bundle.

3. $a_R$ is $\mathcal{G}$ invariant.

4. the actions of $\mathcal{G}$ and $\mathcal{H}$ commutes.

I am reading [Orbifolds as stacks?][1]

Can some one please confirm if this is what it is. [1]: https://arxiv.org/abs/0806.4160

I am reading Orbifolds as stacks?

Can some one please confirm if this is what it is.

EDI : STACKY LIE GROUPS says in page $6$ that

Let $\mathcal{G},\mathcal{H},\mathcal{K}$ be Lie groupoids and $M$ be a $\mathcal{G}-\mathcal{H}$ bibundle and $N$ be a $\mathcal{H}-\mathcal{K}$ bibundle. Viewing a bibundle as relation of stacks suggests defining the composition of bibundles as $M\circ N=(M\times_{\mathcal{H}_0}N)/\mathcal{H}_1$ where the quotient is with respect to diagonal action $(m,n).h=(m.h,h^{-1}.n)$ for $m,n$ wherever defined.

I am not aware of seeing bibundles as relation of stacks. Can some one help me to see bibundles as relation of stacks. May be then, It would be easier for me to understand why that choice would natural when defining composition.