Understanding the definition of $G$-gerbe

Question

In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following.

Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\rightarrow \mathcal{C}$ which locally is isomorphic to $[pt/G]\times \mathcal{C}$.

I am not able to understand what is locally isomorphic here.

Can some one help to clarify this.

I am trying to make sense of this by imitating what does it mean to say principal $S^1$ bundle over a manifold. It means a smooth map $P\rightarrow M$ that locally looks like product i.e., there exists an open cover $\{U_\alpha\}$ of $M$ with trivializations $\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times S^1$. One can see this $\pi^{-1}(U_\alpha)$ as pull back of inclusion $U_\alpha\rightarrow U$ along $\pi:P\rightarrow M$ .

If we imitatie, by locally isomorphic we mean, I think it means there exists an atlas (open cover in above sense) $\underline{X}\rightarrow \mathcal{C}$ such that the fiber product $\mathcal{D}\times_{\mathcal{C}}\underline{X}$ (pull back $\pi^{-1}(U_\alpha)$ in above sense) is some how related to $[pt/G]\times \mathcal{C}$.

In Differentiable Stacks and Gerbes Kai Behrend and Ping Xu defines an $S^1$-gerbe as the following.

An $S^1$-gerbe over $\mathfrak{X}$ is a gerbe $\mathfrak{R}\rightarrow \mathfrak{X}$ which is locally isomorphic to $BS^1\times \mathfrak{X}$ and is endowed with a trivialization of its band (the $2$-sheeted covering $\underline{Band}(\mathfrak{R})\rightarrow \mathfrak{X}$).

In this also there was not much explanation of what is locally isomorphic to.

Any comments on definition of Band is welcome.

In Some notes on Differentiable stacks J. Heinloth defines a $G$-gerbe as the following.

A gerbe $\mathcal{D}\rightarrow \mathcal{C}$ is called an $S^1$-gerbe if there is an atlas $\underline{X}\rightarrow \mathcal{C}$ and a section $s:\underline{X}\rightarrow \mathcal{D}$ such that there is an isomorphism $(X\times_{\mathcal{D}}X)\times_{X\times_{\mathcal{C}}X}X\cong S^1\times X$ "as a family of groups over $X$" with some other conditions.

By specifying "as a family of groups on $X$" I think he want to see $S^1\times X$ as not just like a manifold but see $S^1$ as a Lie group and $X$ as a manifold separately I mean the stack associated to $S^1\times X$ as $BS^1\times \underline{X}$ and not $\underline{S^1\times X}$. This seems compatible with what Gregory says i.e., locally isomorphic to $[pt/G]\times \mathcal{C}$. Here $G=S^1$ and $[pr/G]\times \mathcal{C}$ is $[pt/S^1]\times \mathcal{C}$ i.e., $BS^1\times \mathcal{C}$.

Can some one help to clarify this.

Answer 1

In Brauer Groups and Quotient stacks, they define $G$-gerbe as follows.

Set up : Fix a Noeth. scheme $X$. Let $G$ be a group scheme (flat, separated and of finite type) over $X$.

A $G$-gerbe over $X$ is a morphism $F\rightarrow X$, with $F$ an algebraic stack, such that there exists a faithfully flat map, locally of finite presentation, $X'\rightarrow X$ such that $F\times_XX'\cong BG\times_XX'$.

As $G$ is a group scheme over $X$, there is an obvious morphism $BG\rightarrow X$. So, we can then talk about the product $BG\times_XX'$. This may not be the case in case of differential geometric set up.

---

A morphism of stacks $F:\mathcal{D}\rightarrow \mathcal{C}$ is a $G$-gerbe, If it is a gerbe over stack in the usual sense and, I think what they mean is, there exists an atlas $p:\underline{X}\rightarrow \mathcal{C}$ such that the pullback $\underline{X}\times_{\mathcal{C}}\mathcal{D}$ is isomorphic to the stack $BG\times_{\text{Man}}\underline{X}$.

---

I also think there should be something like an action map $BG\times_{\text {Man}}\mathcal{D}\rightarrow \mathcal{D}$ which should be behaving like Lie group action on a manifold $P\times G\rightarrow P$ in case of Principal $G$ bundle.