I am trying to figure out the conditions under which you can glue together a collection of (differentiable) stacks by equivalences, and get a differentiable stack.


More precisely, I have a collection of stacks $U\_i$ and open substacks $U\_i|\_j\hookrightarrow U\_i$ with equivalences $\varphi\_{ij}:U\_i|\_j\to U\_j|\_i$ so that $\varphi\_{ij}\circ \varphi\_{ji}\cong \operatorname{Id}\_{U\_i|\_j}$. I want to know when and how this determines a differentiable stack.


If we assume the $U\_i$'s are manifolds and the $\varphi\_{ij}$'s satisfy the cocycle condition $\varphi\_{jk}\circ \varphi\_{ij}=\varphi\_{ik}$ (after restricting the domains), then gluing along the $\varphi\_{ij}$'s will give us a (possibly non-Hausdorff) manifold $M$. We can think of $M$ as the coequalizer of the groupoid $\coprod\_{ij} U\_i|\_j\rightrightarrows\coprod\_i U\_i$ where source map is inclusion, target is $\varphi\_{ij}$ composed with inclusion, etc. (This is just a reverse-engineered Cech groupoid)


My hope/guess is that the picture for stacks works roughly this way, that maybe if we have some weak cocycle condition ($\varphi\_{jk}\circ \varphi\_{ij}\cong\varphi\_{ik}$), maybe we can glue everything together as a (weak?) coequalizer of the appropriate groupoid (which is now a groupoid internal to the category of stacks). I'm not sure if this is right, and even if it is I don't really know how to show it.


Any thoughts/references would be much appreciated, thanks!



Gluing together together differentiable stacks
Try p. 17 of notes by Breen (<http://math.uchicago.edu/~may/IMA/Breen.pdf>) *Notes on 1- and 2-gerbes*


