# Differential gerbes as groupoid extensions

I started reading Non abelian differential gerbes https://arxiv.org/abs/math/0511696v5

It says in abstract :

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids.

It says in section 2, named Differentiable gerbes as groupoid extensions that,

The purpose of this section is to set up the basic notions of differentiable gerbes in terms of Lie groupoid extensions.

Then they start discussing what is a Lie groupoid extension and all but I was not able to find any definition for what does it mean to say differentiable gerbe or how they are relating differentiable gerbes with Lie groupoid extensions.

There is only one remark in page $8$ saying

There is a 1-1 correspondence between Morita equivalence classes of Lie groupoid extensions and (equivalence classes of) differentiable gerbes over stacks.

But I do not see any reference given for this remark.

Can some one help me to understand how differentiable gerbes(I know them as stacks over a manifold that are locally non empty and locally connected) are related to Lie groupoid extensions.

More over, what does it mean to say differentiable gerbes over stacks? Does it mean something to do with morphism of stacks where domain is a gerbe?

Are there other places where Lie groupoid extensions are discussed in detail?

Question : Can some one provide a reference for seeing Differentiable gerbes on stacks as Lie groupoid extensions.

I remember seeing gerbes on Manifolds are some how related to some crossed modules extensions. Could not recall or find the reference for that as well.

• May I suggest that, before you keep looking at this line of questioning trying to understand the general case, you read up using anything from the extensive literature on bundle gerbes. You could do worse than start with this review: arxiv.org/abs/0712.1651 – David Roberts Jul 8 '18 at 8:07
• @DavidRoberts Thank you for your response.. As suggested by you, I will start reading that review... I some how enjoy reading gerbes as stacks of certain kind... I will definitely tell you how I understand that article of bundle gerbes... I believe you never know if some thing would be joyful/useful unless you read it :D – Praphulla Koushik Jul 8 '18 at 12:21
• If you haven't seen it, this may also be useful: arxiv.org/abs/math/0503247 – David Roberts Jul 11 '18 at 23:20
• @DavidRoberts I have not seen that before. I skim through notes but could not find anything related to Lie groupoid extensions relation with gerbes.. is there anything specific you want me to look there? It looks good though.. I will read it anyways... – Praphulla Koushik Jul 12 '18 at 1:33
• Lie groupoids have underlying topological groupoids, so there is a lot you can learn by starting with topological groupoids. Then working with Lie groupoids you then have to learn how the extra properties (source and target being submersions) make a difference. – David Roberts Jul 12 '18 at 7:05